The ability of colleges/universities to make their own rules regarding guns is derived from § 15.2-915. wherein only municipalities are restricted by preemption.
Thereafter the Va. Administrative Code leaves such schools of higher learning the ability to make rules. These rules may restrict students and employees from doing certain things which frequently include disallowing guns under potential penalty of being expelled or fired.
Private citizens not affiliated with the school are not however so governed. The recent GMU case (DiGiacinto v. The Rectors and Visitors of George Mason University) was decided in favor of GMU insofar as students and staff + for others inside the buildings. The SC did not support that all guns be banned from the university, leaving the non-affiliated the otherwise legal ability to CC or OC on the grounds thereof.
Private colleges, not accepting any public funds (are there any?), would still be able to restrict any and all. Regarding all universities in Va., the only law that might be broken would be that of trespass - there is no more severe penalty.
As a foot note, I have carried at William & Mary, Univ. of Va. as well as numerous community colleges throughout the state.