Foreign students would not necessarily have a social security number, alien card, etc. but would be permitted to attend classes if they were in the USA on a student visa and plan to pursue their education here and return to their country of residency thereafter. Can we enroll those students in our school programs? Is there some other permission document that they would need to acquire prior to enrollment?

I have experienced that whether you speak directly or indirectly with a prospective student, that the best way to make any corrections on information that was subsequently given out to them, is to do it in writing. This way, you have a paper trail of the corrective measure that was taken to ensure that the prospective student was accurately informed.