Fax machines have a phone built in, and it expects to be connected to the normal phone system. I don't know of any fax machines with a SIM-card, let alone fax machines that would communicate wirelessly with a cell phone to use that to connect.

And I know of no cell phone that lets you connect to a second local source like a fax machine.

It might well be, by the way, that the wireless network doesn't like the fax (or modem) beeps either. So I'd say it's a two or three way no.

Cell phones DO support SMS for text. And smartphones support e-mail. But that isn't what you want, probably.