The question is not and has never been "are spiders immune to poison?"
It is "is stabbing a swarm of spiders with a poison-coated rapier an effective means of using said poison to poison them?"
Obviously, this would be a terrible way of trying to poison said spiders in the real world. In the real world, you would try to use some sort of area affect poison like an aerosol spray or poured liquid.
To those arguing that what the rapier wielder is REALLY doing is not stabbing, but whipping the rapier around and splashing poison on the spiders -- no. The rapier is doing PIERCING damage, and unless the poison being used is specifically a contact (as opposed to injected) poison, it is poisoning the spiders not by touching them but rather through entering their bodies via a wound.
You can defend game-world vs real-world logic, physics, and whatnot, but the reality is that stabbing a swarm of spiders with a poisoned rapier is an effective way of killing them ACCORDING TO THE RULES because 5E simplified swarms to the point of frequent silliness.
Are you OK with that silliness? That is a question for you and your table to answer.
I would add, though, that all too often a table of PLAYERS will balk at following the rules off of a silly cliff when doing so would be disadvantageous to their characters, while also digging in their heels and insisting that the DM follow the rules to the letter (no matter how silly the outcome) when doing so would be advantageous to their characters. This is how things often were back in 2E and before, and why many DMs were all too happy to see rules clearly spelled out to an almost microscopic level in 3E -- before they realized what splatbook proliferation and CharOp forums would do with those detailed rules (remember, back when 3.0 first came out, internet gaming forums were still limited to Usenet's rec.games.frp.dnd and the like).
Now that the rules are more fluid and based on DM call, the name of the powergame is not (so much) to rules lawyer but rather to badger, cajole, and in other ways convince the DM to rule how you want them to rule, even in ways contradictory to the way you wanted him to rule fifteen minutes ago.
As I have said before:
5e did not eliminate powergaming. It just changed it from an Int-based skill to a Cha-based one.