Possible to make flask login system which doesn't use client-side session/cookie?
Nope.  That's literally impossible over pure HTTP.  HTTP is a stateless protocol, which means that in order to preserve state, the client has to be able to identify itself on every request.
What you might be able to do is HTTP Basic Authentication over HTTPS, then access that Authentication on the server side.
I'm working on a Flask app to be used by a medical client. Their IT dept is so up tight about security that they disable cookies and scripting network-wide. 
Luckly, wtf-forms was able to address one of these issues with server-side validation of form input. 
However, I'm getting hung up on the login system. I've implemented flask-login, but this apparently requires client-side data as I'm unable to log in when testing in a browser with these features disabled. 
Is there any way to create a login with zero client-side data? 
Thanks for the help.