Note that often we use електронна пошта (well, electronic mail :))
I agree, as a native Ukrainian. We tend to use "email" when we are talking, but I have never seen it written like "імейл". We use "електронна пошта" instead, or just English "email" - it's quite acceptable.
I think that in English, careful writers would phrase the question as "What is your email address?" Is it similar in Ukrainian? Would writers typically include, e.g., "адреса"?
Sure: Яка ваша адреса електронної пошти? Or just: Яка ваша електронна адреса?
I agree that in English we would usually say email address, but perhaps that is changing. Some people already use email when they mean email address.