Password Reset from email link stopped working

All of a sudden, attempts to reset passwords with the “forgot your password” email stopped working:
On Jan 2, someone successfully used the functionality, on my site, but now, after receiving the email and using the link and picking a new password, my users are getting only a blank page with a little bit of text:

{“success”:false,”data”:{“message”:”Password is missing!”,”for”:”password1″}}

I have never seen this and have not been able to figure out what would be throwing such an error.

Also, I had not done any updates to any plugins on my site (or any other changes) between Jan 2 and 22 (when I was first made aware of the problem).

The page I need help with: [log in to see the link]