Applying poison to a weapon is an action. Can I use the familiar's action to apply poison to a weapon I'm holding?
Please cite only the rules and those rules only of the PHB to support your answer not with guesswork
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Find familiar states:
A familiar can't attack, but it can take other actions as normal.
One of the actions listed as an example for "the sorts of thing you can do in tandem with your movement and action" is:
withdraw a potion from your backpack.
Since a familiar may "take other actions as normal", I would rule that a familiar may use its free object interaction to withdraw the poison from your backpack, and use its action to apply the poison to your weapon.
MivaScott helpfully pointed out in comments the the scenario described above may be a bit dubious with respect to action economy. I offer an alternative scenario, concluding still that a familiar may use its action to apply the poison.
Conceivably, a PC with a single free hand could on its turn withdraw the vial of poison from the pack. The familiar could then (arguably) use its interact object to retrieve the vial from the PC's hand, and use its action to apply the poison. Taking an item from another character is not mentioned in the interact object list, but its a list of examples not meant to be exclusive.
Note that there’s nothing that inherently stop this from working initially – a familiar can perform actions, and applying poison to a weapon is certainly an action.
However you put it though, an Action needs to be something that the creature taking that action is physically capable of doing. There are of course no explicit rules for what an animal can or can't do, but this is not unexpected: For example, the limitations of general interaction with the physical environment aren't explicitly encoded anywhere in the rules, rather the rules assume you're roughly familiar with what humanoid things can and can't do and expect you to act and your DM to rule accordingly. The general rule for this is actually in the start in the How to play section:
So where is the problem here?
Well, the problem is the forms your familiar can take:
bat, cat, crab, frog (toad), hawk, Lizard, Octopus, owl, Poisonous Snake, fish (quipper), rat, raven, Sea Horse, Spider, or Weasel
(The Pact of the Chain feature adds some options and your DM may offer additional forms at their discretion).
Most of these simply aren’t physically capable of opening a flask or a bottle and applying its contents over a weapon. You could certainly argue some of the more dextrous ones among these such as a raven or an octopus should be able to, but how is a Snake, Sea Horse or Spider going to apply the poison? You’ll at least have to come up with some sort of plausible description for these things.