"Email me" and "mail to me"

The only person I know who says "mail to me" is very much not a native speaker of English. "Write mail to me" is only marginally better.

Mail me that package.
Email me that report.

Please [write|send] me a letter.
Please [write|send] me an email.

Write me!
Email me!

I got lots of mail.
I got lots of letters.
I got lots of emails.

Nobody ever sends me any mail.
Nobody ever sends me any email.
Nobody ever writes [to] me.
Nobody ever emails me.

Please mail that card to me.
Please email that picture to me.

As you can see, there are some differences between how the word "mail" is used vs. how the word "email" is used, but that difference isn't that "mail" takes "to" while "email" doesn't. It's more that "email" is used as the electronic equivalent of not just "mail", but also "letter".

"Correct"? That's just what many people use. If that makes it "correct", so be it.

Some of us can't generate that construct. We just just say send me mail, always. Well, or mail me, I guess. I never say email at all, really. Mail means email. Snail-mail is postal mail.