You can ask someone to mail you, but should you?
Please mail me the file
doesn't sound right to me. If we usually email each other and don't use postal mail, I would understand what you mean. But, I might be confused for a moment, or stop to think that it sounded awkward.
Even though mail as a noun sometimes means electronic mail, mail as a verb like in mail me usually means postal mail. There are not any examples in the COCA (Corpus of Contemporary American English - a set of recent spoken and written language samples) where mail me seems to mean email.
The use of mail might be changing or might be different for people whose English is not in the samples in COCA. But it is a safe choice to write email instead, as in:
Please email me the file
If you want to choose a different word, send is common, as in send me an email, send her a text, or send us the information. Maybe because I hear and see send so much more than mail, please send me the file sounds normal to me. It would not confuse me (if we usually send emails and not postal mail).