Can one order by using, "Je prends un(e) _____, s'il vous plaît." ? Is it polite enough?

Bonjour Caroline and Chris !

Saying "Je prends une bière, s'il vous plaît." would be equivalent to saying "I'm having a beer, thanks."
I do think it sounds a bit awkward in both languages, as it's both assertive and asking for approval (please) :)
But as Chris said, I'm sure it can be used by some people, though personally I would say "Je vais prendre une bière" or "Je prendrai une bière" in that context, which sound less imposing in my opinion.

I hope that's helpful!
Bonne journée !