I searched but maybe I'm using the wrong terms, but as above can I use a fax machine with a VOIP phone?
That is incorrect. A properly operating VoIP link is indistinguishable from a regular TDM digital voice channel. However, the question remains whether you get good enough VoIP over the public Internet. I have set up FAX over VoIP, a few times, and it worked well. There is even a FAX CODEC that's designed to work over VoIPFAX was actually designed for analog network and cannot travel on VoIP network