Does "Poehler claims her Pudding Pot" mean...?

The "pudding pot" is simply a trophy for a humorous award. It is all a joke that is part of a dinner event. The word "claim" in this context simply means "accept".

She was given a joke award which she accepted.

Do you mean that "pudding pot" is just a funny story she received as a award?

The "Pudding Pot" is an award that is given out by a Harvard student club. It is all done in humor, they invite the award winner (generally a celebrity) and then "roast" them (make jokes at their expense). The evening is good fun for all.

The award consists of an actual trophy in the form of a pudding pot.

Apparently, it's a real award.