Minor note for my aproach: I treat the team like it's an in game boss, so that's why I mention the AI and only give an item to the final Pokemon.
Other than the 3 Poison Types, I'll start with who learns Poison moves without TMs. If we get to who learns Poison moves through TMs, I'm ignoring Toxic. That'd let me use anything except Tynamo. I'm also ignoring anyone that only learns Gunk Shot though Tutor because that wasn't until BW2.
Accelgor learns a Poison Attack by Level, Acid Spray at Level 4, and Shelmet learns Acid at Level 4 instead as well as Acid Armor. This slimy shelless snail is good thematically, but not very good when there's already another Grass type and another Bug. Then again, this is more about representing the type than making a good team...
Cottonee learns Poison Powder. Honestly, it feels like a Grass move that's Poison instead because it poisons. Still a little tempting because of how much I love Leech Seed though. :P
Seismitoad learns Acid thought Level and has the Ability Poison Touch.
Galvantula only has Gastro Acid through Level, but has the Egg Moves Poison Sting and, much more usefully, Cross Poison.
Eelektrik learns Acid, Acid Spray, Coil, and Gastro Acid. It doesn't learn Sludge Bomb and the AI won't use Acid Spray, so Coil will do for Eelektross!
Druddigon has Poison Tail as an Egg Move. Interesting.
Chandelure learns Smog and has 3 Poison Egg Moves, Acid Armor, Acid, and Clear Smog. None of those are good enough. That's a shame because I would've liked to use a Ghost Type, especialy because of Hex.
Serperior learns Coil and Gatro Acid, but no one in BW uses starters other than you and your rivals.
I don't even need the no starters reasoning to say no to Emboar's Smog.
Garbodor
Aftermath
-Toxic Spikes
-Gunk Shot
-Payback
-Toxic
No Poison type specialists in game (except the one that cheats and sets it up before the battle) uses Toxic Spikes. Let's fix that! A lead with Toxic Spikes forces you to deal with poison, which is kind of the main point of Poison. Payback is one of the few coverage moves and happens to hit Psychic Types Super Effectively. The Movepool is bad enough that I considered Explosion, but the risk of the AI using it Turn 1 isn't worth it. Body Slam's 30% chance to inflict a status problem that isn't poison also isn't worth it because it means no poisoning. I just slapped on Toxic for the fourth move. Even though no one in game used Hidden Abilities until SwSh, Gen 5 did introduce them. I'd say that makes up for not having any moves that were new in BW.
Note edited in after noticing most other Garbodor are Special: The reason my Garbodor is Physical is their base Attack is 95 and their base Special Attack is 60. The Special Movepool is better so I probably should have ignored the stat difference. It's not like one stat being so much higher hasn't stopped Game Freak from making movesets that use the lower one.
Seismitoad
Poison Touch
-Sludge Wave
-Muddy Water
-Dig
-Drain Punch
Sludge Wave is slightly more powerful than Sludge Bomb, much fewer Pokemon can learn it, and it's a new move in BW. Muddy Water has the same Power but worse Accuracy than Surf, but if feels right to have the dirtier move for a Poison team. It's not completely worse than Surf though because of the 30% Accuracy drop chance. I'm not going to deny Dig isn't the best Ground Attack, but it's Seismitoad's only STAB the makes contact and what's the point of Poison Touch if you're not using contact moves? Drain Punch is a contact move that's coverage for... no idea honestly. I just think it's odd that Seismitoad learns it by Level. Pokmeon in game have random covorage moves all the time and it's much better than the Fighting Elite Four's Grass Knot on Sawk. (I have no idea what Game Freak was thinking when they made that moveset or what the AI was thinking when they used it on my Servine.)
My reason for Seismitoad, other than Poison Touch, is the mud of Water/Ground types is easily associated with the muck of Poison types. Swampert and Gastrodon even have Sludge as Egg Moves!
Accelgor
Hydration
-Sludge Bomb
-Acid Spray
-Bug Buzz
-Focus Blast
Acid Spray's effect would be threatening if the AI would ever use the 40 Power move. Sludge Bomb is so there's a Poison move the AI would actually use. Bug STAB and Focus Blast coverage are unplesent for Psychic and Steel types. I could've used Energy Ball or Giga Drain for Accuracy and hitting Ground Types, but the only Pokemon weak to Ground on this team is Garbodor.
My reason for using Accelgor, even though its weaknesses are terribble for a team that already had a Grass type and another Bug type, is Shelmet is only found in the swamps of Route 8 and the Moor of Iccirus. Swampy and slimy just feels right to me for Poison.
Eelektross
Levitate
-Coil
-Toxic
-Wild Charge
-Brick Break
Coil is the slimy eel's obligatory non-Toxic Poison Move and Toxic is for for a Poison move that's more obviously a Poison move and works well with Coil's Defense boost. Wild Charge's recoil makes it a bad choice for a setup sweeper, but it's better than Spark and can give players the realization that they just need to endure long enough for Eelektross to KO itself. I wanted to have a coverage move that's resisted by Grass to throw a bone to the only type weak to Poison, but there's weren't any. Instead, I went with the move that's a screw you to the Grass Type that gets off too easy, Ferrothorn.
I think Eelektross is the least Posion Pokemon on this team, but I still liked it better than the other Pokemon that lean Poison moves naturaly because they mostly learn usable Poison moves as Egg Moves and eels are slimy. It also gets to show off a Poison Status move that doesn't inflict poison.
Amoonguss
Effect Spore
-Toxic
-Venoshock
-Giga Drain
-Synthesis
There's no way Amoonguss can't not have a stall moveset. Just because we have Toxic Spikes doesn't mean we don't have to use Toxic. Flying Types ignore it and Poison Types absorb it after all. Venoshock is used over Sludge Bomb because it's more powerful after poisoning the opponent and it's a "new" move. The coverage isn't good, but coverage isn't really an option for Amoongus.
Scolipede @ Black Sludge
Poison Point
-Poison Jab
-Megahorn
-Earthquake
-Sword Dance
Amoonguss's movepool isn't good enough for a tough ace and Garbodor is the lead for Toxic Spikes, so Scolipede gets to be the ace. Sword Dance on a fast Pokemon is terrifying, especially with Megahorn's 120 Power, but the 85 Accuracy gives the player hope. Earthquake hits Fire types, which 3 Pokemon on the team are weak to, and the types that are immune to poison. I considered Bulldoze or Steamroller for the sake of having a "new" move, but nerfing what's supposed to be the team's signature Pokemon didn't feel worth it for more representation of the "new" generation. I also considered a Poison Gem instead of the usual healing berry because it's Gen 5 only, but Black Sludge is practically signature item of Poison types. I don't think I need to explain the Ability.
Pretty weak to Fire and Flying, but I think it's strong thematicly and that's more importaint Coincidentally, the only Poison move more than one Pokmeon on this team knows is Toxic. That wasn't planned but it's a nice bonus for showing off diferent Poison moves.