That's not exactly complicated if either party owns a web server. Which - last I checked - the government has.
Just give the person who needs to send the sensitive documents a short link like uploaddocuments.gov, have that page ask for some basic identifying info, and have a box for the user to drag and drop a file. At which point the browser will p2p upload that file over HTTPS.
I don’t love faxes. This isn’t me saying we should keep them forever. We shouldn’t. Still, there are reasons they’re still widely used for medical stuff today. If CMS or HHS rolled out a new method and told doctor’s offices to start using it if they want to get paid, the industry would switch in a heartbeat. Short of that, any other alternative will take approximately forever.