At UVa, VT, and the other public colleges which are open for citizens carry they would need probable cause for any criminal code action -- the same as in any town county, etc.
OTOH, when a students enroll, or an employees are hired, they enter into a contract in which they consent to all types of restrictions and agree to the lose of rights relative to administrative and civil matters -- be it guns, drinking, or shorts.
Private colleges' restrictions are backed up by trespass violations. I'm not at all sure how search of non-students/employees would work. OTOH, the contract with students/employees would likely be binding, for the college to search at private colleges.
In any case, expelling someone is not a criminal matter and none of the Bill of Rights play. "You have a right to [fill in the blank], but you do not have a Constitutional Right to be a student/employee/on campus/etc.
It is not unlike lots of employers which have no-gun polices and can legally search your car in their parking lot.