Specifically, I'm looking for an answer to the first effect of the Poisoner feat:
"When you make a damage roll that deals poison damage, it ignores
resistance to poison damage." Tasha's Cauldron of Everything (p. 80)
My question is: for a wildshape form that has innate poison like a poisonous snake, or a giant spider, does this effect apply to an attack made in that wildshape form?
This quote from the Player's Handbook (p. 67) also comes to mind:
You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so.
So naturally, it seems the question becomes if it counts as "physically able" to use natural poisons like artificially refined poisons.
Best Answer
You keep the independent benefits of the feat in wildshape form
As you correctly indentified, the feat is a feature granted by your class (or possibly race if you are a variant human), and will be available to your new form if it is physically capable of using it.
The Sage Advice Compendium in an answer to a question about Crossbow Expert made it clear that each bullet of the abilities granted by a feat should be seen in isolation:
(In that case, being an expert in crossbows helped ranged spell attacks, which have nothing to do with crossbows, even though the feats introduction text says "Thanks to extensive practice with the crossbow, you gain the following benefits".)
As a consequence, when you attack with a poison bite or sting in your shapechanged form, the poison damage caused will ignore resistance.
I still would check with my DM: the framing language of Poisoner states:
Your DM might not agree with the rulings in the SAC, and rule that due to this context, the benefits only apply to poisons that you did prepare, not to natural poisons.