How much possibility does 'may' mean?

Without more context it is hard to say. Strictly speaking "may" does not imply any probability other than the possibility, but that is not always the case. For example the sentence

You may leave your umbrella by the door

Is a polite way of saying "Do leave your umbrella by the door."

In other contexts "may" may be used instead of "will" or may mean there is an expectation. For example, a brochure for a summer camp may read

Young campers may explore the great outdoors...

Which brings me back to your sentence. It could mean that young patients happen upon potential future disease or it could mean that patients are placed in a situation where the expectation is that they will learn of potential future diseases. Placing your sentence in more context as an example:

The support group offers advice and guidance where young patients may learn what diseases they could get as they get older.

Like with the camp example, something is offered with the expectation that the offer will be accepted.

According to Webster's 4th ed:

"Possibility - the quality or condition of being possible"

"Possible - that can be; capable of existing"

This is an on or off word. Something is either possible or not. "May" connotes that it is possible, but conveys nothing of "will" or "likely-hood" or "probability" - which was probably the word you are looking for in "How much possibility..."

"May" in this context conveys that it is possible or that they will have the option if they so choose, or should circumstances align in a way that they do.

Replacing "may" with "shall" or "will" implies 100% probability that this will occur. To say "will likely" instead actually implies a probability somewhere greater than 50%, or to say "will not likely" somewhere less than 50%.

In short: "may" does not imply any probability, but declares it is possible. Additionally, possibility has no metrics other than on or off, yes or no.