There are several issues with this proposed regulation... 

Hypocrisy:  this regulation now tells me that I cant carry a concealed weapon in buildings controlled by the very same government that issued me the concealed carry permit.   The state police performed the background check, why does the Governor not have faith in the concealed carry permit program?  

Nonstarter:  the only people that will obey this law are law abiding citizens, those that will carry without a permit are already breaking the concealed carry laws.  There is no reason to create more regulation and law when it will do nothing to prevent the perceived problem.  

Inconsistent:  the text of this regulation is not consistent in the definition of terms used.  The definition of firearm differs from other parts of the code of Virginia see 18.2-308.2:2 for one example.  Other definitions differ as well. 

Not a legal use of power:  The authority granted to the governor by this provision was not intended to be used for gun laws.  https://vacode.org/2.2-1102/ is rather vague verbatum, however a search of its intended purpose shows some of the amendment attempts reference procurement contracts, car rentals and surplus sales.  If this regulation is believed to be legitimate, then allow this to be voted on by the generally assembly.