Is there any reason a pc cannot gain a creature's action card (like from the creature vault) as an advancement besides theme and the gm's judgement?

I don't see any reason why not. But then 'GM's judgement' covers pretty much everything.

There's no assurance of play balance then so it's "void the warranty on fun, you're on your own for how that works out" - maybe fine, maybe not. If do try it, let us know how it works out.

I've actually been wrestling with this one. My players were helping me sleeve some cards, and our noble asked to buy a card that is essentially a dodge/parry/block for social maneuvres. I honestly am thinking of letting him have it. It's very appropriate but I'm of course worried about precedent.