"May I?" by itself is ambiguous without very clear context. "May I get <X>?" would be arguably less incorrect than "Can I get <X>?" but it would still be completely incorrect because 'get' is completely the wrong word in that context.
Using 'get' means that your are asking if you can fetch X yourself. You as the person doing the action of getting X. If that's what you mean, it's correct. If it's not what you mean, it's incorrect.
So, for example, if you go up to a counter and ask the person behind it "Can I get a latte?" then you're asking them if you can go behind the counter and use the coffee machine yourself. Which probably isn't what you meant, so it's probably incorrect. If you want someone else to get <X> and give it to you, the right word is 'have', not 'get'.
Could/may is something else. Strictly speaking, 'could/can' is about if you're able to do something and 'may' is asking for permission to do it. But that distinction isn't made so much nowadays. On the same basis, you could argue that "could I get?" is now right despite being obviously wrong simply because enough people are using it incorrectly. Like 'literally' now also meaning 'the complete opposite of literally' and 'unique' now also meaning 'definitely not unique but maybe not very common'. Neither make any sense and both are obviously completely wrong and go directly against the main point of language (communication), but from a descriptivist point of view if enough people are doing it wrong then it's right.