FOLLOW-UP
This problem returned after not having it for a month or so, and here's what fixed it this time.
It was the server. I went into the WHM part of cPanel and restarted this function:
PHP-FPM service for Apache
I found this solution by looking back through our service requests on our server provider, to see what they had done the last time.
At present, we are able to restore again without receiving an error message like "missing parameter" or "PHP is missing a folder." Those error messages apparently do not report the problem or point toward the solution.
It is possible that rebooting the server would also have fixed it. If the problem recurs, I'll try that.