Mildly off topic, and showing my age a bit on this one but it 100% is easy, as long as the telco in question supports it.
I used to be with Koodo up here in Canada (albeit this was years ago) and the ability to send/receive emails via your phone number was trivial. Literally could email 5555555555[AT]msg.koodomobile.com and it would be received, and when you reply the other side gets the email reply.
Whether companies still do this on the other hand, is the question (that no one asked lol).