This build formerly used Dispensary as a recommended subsume. While it is still pretty good, as of Lua's Prey, Nourish is a very good subsume for energy gain, Viral damage boosting [Flechette] Orbs and Viral damage staggers, so it is recommended in place (but you should always try them out for yourself).

IMPORTANT NOTE: For people new to the game, "Bastille" was formerly 2 separate abilities: "Bastille" (tap functionality, the enemy suspension field) and "Vortex" (hold functionality, the self-explanatory sucking [Vacuum]). This guide will refer to them as Bastille and Vortex as such. I also expect you to have a basic understanding of Vauban's abilities if you're looking for a build for him. Please at least read what the wiki says about them. https://warframe.fandom.com/wiki/Vauban/Abilities

This is a [Vauban] build. It focuses on using his 2nd (mostly [Flechette], but Tether Coil and Overdriver are nice too), 3rd (which is infused over) and 4th abilities (mostly Vortex, but Bastille is nice situationally). You can use normal [Vauban] or [Vauban Prime] for this, it's whatever.

Power stat prioritisation:
Range is the most important stat. Longer tether coil pulling, longer Bastille and Vortex range.
Strength is the next most important. Dealing more damage with Flechettes is always a plus. Some would argue that this is "unneeded" because [Flechette] orbs already scale with enemy level, but bear in mind that 200% strength versus 100% strength is still dealing double the damage for example. You will still kill slower unless the enemy would be oneshot regardless.
Duration is still important, but it's fine to not have it that high. Just recast your abilities.
Efficiency can be dumped. This example build uses Nourish infused for both the viral damage boost of [Flechette] Orbs (see later) and the amplification of incoming energy. [Equilibrium] is flexible, but used for any health orbs being taken (eg: from [Synth Deconstruct], Eximus kills, etc). Note that [Equilibrium] also helps Dispensary builds a ton, but the energy gain isn't boosted by Nourish. IF YOU DON'T HAVE HELMINTH AVAILABLE, REPLACE [Equilibrium] WITH [Streamline] (or whatever other mod suits your fancy; it is your build).
Additional note: This is a small consideration as we do not use the skill much (also, other armour strip exists), but the strength value for Bastille's armour strip matters. The armour strip happens for each second, so for example having greater than 167% strength (fullstrip in 6s) but less than 200% strength (fullstrip in 5s) will still make you strip in 6s exactly. That being said, here are some power strength thresholds to take note of if using bastille (this build strips in 6s without outside sources)
167% - fullstrip in 6s (for this reason, you are able to have r9 blind rage instead of r10, but it's not too too important)
200% - fullstrip in 5s
250% - fullstrip in 4s
334% - fullstrip in 3s

Buffing [Flechette] Orbs:
If you have not read the wiki yet (which you should), here is a recap of Flechette orbs:
It scales off Power Strength, Vauban's Passive, ceil(Enemy Level/10) as well as Weapon Buffs.
Power Strength is already covered earlier. So the following will consider possible Weapon Buffs.
For the +damage stat, you can use [Vigorous Swap] or Arcane Arachne. While you can use both, I would not really recommend it because they are additive. You can save a mod or arcane slot for something else. In addition, the new [Holster Amp] aura can also buff [Flechette] orbs. This is just a tiny bonus compared to the aforementioned two though.
For +critical chance, Arcane Avenger works in buffing the orbs. [Smeeta Kavat]'s crit buff also lets [Flechette] orange crit, but curiously [Adarza Kavat] does not work on [Flechette] Orbs.
For +critical damage (which this build is not currently buffing), [Tek Collateral]. This requires standing in the Kavat zone though. Annoying if not camping.
For +elemental, we are using Nourish to add Viral damage to our flechettes. While there is the potential for Viral procs, they don't happen often enough to be considerable.
Other Helminth infusions can buff [Flechette] orbs. Most notably Eclipse (for being a huge multiplicative damage boost while in the light). More info underneath.
After all this, you want armour strip to deal with enemy armour, since armour is still a pain for Flechettes to deal with. This is easily achieved in the form of Unairu's Caustic Strike. Note that Bastille does not armour strip enemies stuck in a Vortex. If using Bastille to strip armour, an alternate grouping source that does not ragdoll enemies is recommended.
Honourable mention: Mutalist Quanta alt-fire orbs buffs [Flechette] nails passing through. Unfortunately the weapon is so jank that I fear that trying to explain it here would get details incorrect. Just know that it buffs Flechette orbs in case you want to power them up even more. Here is the wiki page for it: https://warframe.fandom.com/wiki/Mutalist_Quanta#Alternate_Fire_Mechanics
Other additional note: Theorem Demulcent used to, but now no longer works.

Why no [Tesla Bank]?
Of course, it is up to you to test it out for yourself, since it is your build to use. But Tesla Nervos and Tesla Bank are godawful garbage.
Tesla Nervos is ridiculously slow, and the little CC it does is already overshadowed by Bastille/Vortex. At least the balls look fun to play with.
[Tesla Bank] on the other hand, is ridiculously impractical in situations outside of extremely controlled environments like the simulacrum:
1. Enemies have to be unarmoured. [Tesla Bank] deals electric damage that doesn't proc. Bastille already strips armour, but you'd need a separate source of pulling if using Bastille as Vortex is not compatible with Bastille.
2. You have to target the correct enemies. It is incredibly difficult to see when Tesla Nervos actually attach to enemies (try if out for yourself, if your eyesight is better than mine). Tesla Nervos being slow and unreliable to attach to things doesn't help it either.
3. %-hp based. So now you have to pray to Void that your balls actually attaches to something like a bombard, and not some small fry like a butcher.
4. Vortex. Why nuke enemies in base 8m range, when Vortex pulls in enemies from base 10m range, without any of the aforementioned downsides? If you can kill one enemy to trigger [Tesla Bank], surely you can kill a whole ball of flesh clustered in Vortex. It's not hard at all.
Once again, you use if you want to experiment. You don't have to follow 100% what I say. That is up to you to decide.

You should put [Archon Stretch] to take advantage of its bonus with Tesla Nervos!
First off, 2 energy per second is pitiful, even when boosted by Nourish. Secondly, you have to use the ability which I just outlined a lot of flaws of in the above paragraph. No.

If you [Hate] Tesla Nervos so much, why didn't you get rid of it?
Tesla Nervos actually stuns enemies separately from the electric proc it applies, meaning it's able to lock sentients in place (who aren't really CC-able by most other means). Also, Photon Strike is pretty redundant with how much we're aiming to buff [Flechette] Orbs here. Also, see below.

Helminth Infusion and Choices:
Photon Strike or Tesla Nervos are the infuse slots. I prefer infusing over photon strike because it's harder to fatfinger 2 3 4 than 1 2 4, but it really doesn't matter.
[Vauban] is fine with a bunch of infusions. This example build uses Nourish for energy gain, Viral staggers and boosted Flechette Orb damage.
Other viable infusions:
Dispensary - This is highlighted due to being the previous recommended Helminth ability before the Nourish rework (Lua's Prey). It is a cheap and simple way of generating lots of energy (mostly through Equilibrium), especially without needing any enemies to be nearby. Quite useful.
Reave - Viral proc'd enemies swirling in a Vortex can be spin-reaved to kill them all. Nice combo and alternate damage source to Flechettes. See https://overframe.gg/build/358113/ for more information.
Roar, Eclipse, other weapons buffs (with Streamline over Equilibrium) - These apply to your Flechettes too and are nice dmg buffs. DO NOT use things like Smite for [Smite Infusion] because that doesn't apply to weapon abilities. Note that Eclipse is a bigger boost than Roar if there is enough light.
Spectorage (with Spectrosiphon over Equilibrium) - Alternative energy generation that also works for your team, ideally have Arcane Energize though because starting the combo with 50%/45% efficiency and low starting energy will hurt.
Terrify - Alternative armour-stripping option, if you do not wish to run Unairu. Needs less strength than other stripping options, and the fear does not matter much when everything is stuck inside Vortices.

Shield Gate Note:
This build focuses on shield gate abuse (and insane CC) to survive. Just make sure that at least when you cast Bastille/Vortex it fully replenishes your shields. Your shields will only partially replenish if casting other abilities, but this is not much of an issue due to aforementioned insane CC.
This build uses [Brief Respite] for the full shield gate reset, but you may want to run other auras. In that case, for this 50% efficiency example with Decaying Dragon Key, you'd need 2 Augur mods for a full shield gate with Bastille/Vortex.
Normal [Vauban] has lower shields. With a decaying key, you'd have 56.25 shields instead of Vauban Prime's 75. You'd need only one augur mod for full shieldgating with 50% efficiency with Bastille/Vortex.
Normal [Vauban] is technically "better" in this one case (see below note on starting energy). A very negligible better though. It's not hard to not die. And he looks terrible compared to the prime.

Aura and Exilus:
Both preferential. But (unranked) Combat Discipline is very good in this case as you do not need to get damaged by enemies to trigger Arcane Avenger (and other self-damage arcanes) and Nourish's Viral staggers. This is considerable because you may not be hit much at all, when enemies will be stuck in vortices or getting ball tortured to death. Additionally, it provides constant health damage to be able to pick up health orbs from Dispensary, in the event your companion dies. Do note that a lot more Forma, including an Umbral, is required to fit an unranked Combat Discipline - feel free to use a higher ranked [Combat Discipline], although the health drain will be more severe.
[Holster Amp] is also an option for a little extra damage on Flechettes.
See above note if replacing [Brief Respite].

(Minor) Starting Energy:
This section is for if you're using Dispensary. Otherwise, you can safely ignore it.
Dispensary costs 112.5(116.25) energy with 50%(45%) efficiency respectively.
If you are not already aware, Warframes have +5 base energy per unused mod capacity, and per added capacity (after taking into account polarisation) from the aura mod.
Since Vauban Prime has 100 starting energy, you'll need +3(+4) unused mod capacity and/or added capacity from aura mod to be able to cast dispensary at the start of the mission.
Normal Vauban has 75 starting energy, requiring +5 more unused mod capacity and/or added capacity from aura mod (to the above values) to be able to cast dispensary at the start of the mission.
This example build with [Brief Respite] granting +14 x 5 = +70 starting energy is able to cast Dispensary fine at the start of a mission. Unranked [Combat Discipline] may not, so be careful.
Of course, you could use Preparation if you really wanted to.

For a pure vortex build see https://overframe.gg/build/218837
Can't be bothered to put in all the effort of building this yet? See https://overframe.gg/build/360040/ for some extremely budget build. Obviously this is not recommended if you care about the frame.

Closing note: Bastille and Vortex slow down overguarded eximus and other enemies that cannot be ragdolled.

In honor of senpai's legacy 🙏 they will be missed.