No, that wouldn't sound correct. You can "take a drink" or "take a sip". Or if you are ordering, "I will take a beer please". It's much more common for Americans to use the verb "to have". For instance, "I'll have a beer please" or "I'll take a sip".

I'll take a glass is fine, as in "I'll take a glass of wine." This usage, however, is more common when a friend is offering you a drink, not when you are l ordering in a bar

The French "prendre un verre" is actually "to have a drink" in English. You can say: Let's go and have a drink. Would you like to have a drink with me? I have time for a quick drink. I went for a drink with some friends.

I think you can say like that: can I get a glass of tea or coffee?