Correct, though the container doesn't have to be "locked", merely, "closed".
Also correct that an employment contract may limit possession otherwise, though the penalty is getting fired, not going to jail.
Also note that Virginia law (which I believe pre-empts federal law to the contrary) having to do with possession of firearms on school property gives broader rights than the amendments to the Omnibus Crime Control and Safe Streets Act of 1968 referred to as the "Gun Free School Zones Act" (thank you, Senator Lautenberg, D. of N.J.). That statute prohibits any possession at all within a thousand feet of the outer perimeter of the grounds of any school (K-12, public or private) unless one have a license or permit that requires a background check conducted by state agencies.