I was able to get into it by going to about:preferences#privacy and going to the Cookies and Site Data section and clicking on Manage Data and then removing mercy.net from the list before trying to log in.

I could then type mercy.net into the address bar, then click the Sign In button. This allows me to login to the site after a few redirects. Then click menu button on the top-left and select My Mercy Account which should take you where you want to be.

In order to log back in after restarting the browser or something, I have to do the same process to keep from going to the loop of just showing the login prompt each time.

It's kind of nuts but this workaround seems to work.

Seems like a lot of rigamarole just to get some blood lab results, or whatever Mercy is emailing to me. Is somebody there at Mozilla looking into an easier solution fix? Why is there such hassle with MyMercy and a log in to them?

Yes, it's been reported. They seem to be doing something wrong after accessing their site, as only the first visit works: https://support.mozilla.org/en-US/questions/1579680#answer-1819384

(Since getting in needs a special access, it's hard to investigate.)