I can not think of a way to do this without editing the code in login.pl (or something equivalent).
You could e.g. edit the success_form subroutine in login.pl so that if
the classlist database does not contain the student's email address,
the "Begin Problem sets" button is not displayed and a warning message
is displayed instead. You would have to decide what "contain the
student's email address" means (the simplest thing would be to require
something the looks like an email address).
If possible, it's probably much better to get an initial list of email
addresses from your registrar and add these automatically when you
(hopefully automatically) create a classlist for your course.
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