Translation:A bowl, please.
"A bowl, please" is the best translation for this. It might sound a bit crass in English but it's not in Japanese. Keep in mind too that "can" and "may" (in this context) have different meanings.
can = to be able to may = asking/ seeking permission/ favour
So, "May I have a bowl, please" should be accepted but there is a way to say that in Japanese which is just as common as "ちゃわんをください" If you say "ちゃわんをもらってもいいですか?" It's translated as "Is it good if I receive a bowl?/ May I have a bowl?"
Also, for the "may/can" conundrum. Can is できます。 May is いけます。
Wiktionary says it can be either a teacup or a rice bowl. https://en.wiktionary.org/wiki/%E8%8C%B6%E7%A2%97#Noun_2