First, are you sure it's rubber? Could it be roll roofing instead?

One repairs rubber roofs with pieces of rubber roofing material,
available at roofing suppliers; plus the special adhesive. Very much
like fixing a bike flat tire.

But as for finding the source of the leak, that's often mysterious and
frustrating. If the original rubber has been coated with a paint,
that may be a sign of its age and perhaps previous leaks. There is
eventual aging, you know, that requires re-roofing; or at least, hiring
a roofing co. to repair larger sections of it.

Keep on searching online for more info - I know it's out there. ?ut
first, make sure it is actually rubber. If you can, find out bldg.
repair history from previous owners.