Let's Look at the Law, Not Political Expediency

§ 15.2-915. Control of firearms; applicability to authorities and local governmental agencies.

A. No locality shall adopt or enforce any ordinance, resolution or motion, as permitted by § 15.2-1425, and no agent of such locality shall take any administrative action, governing the purchase, possession, transfer, ownership, carrying, storage or transporting of firearms, ammunition, or components or combination thereof other than those expressly authorized by statute. For purposes of this section, a statute that does not refer to firearms, ammunition, or components or combination thereof, shall not be construed to provide express authorization.
...
The provisions of this section applicable to a locality shall also apply to any authority or to a local governmental entity, including a department or agency, but not including any local or regional jail or juvenile detention facility.
B. Any local ordinance, resolution or motion adopted prior to the effective date of this act governing the purchase, possession, transfer, ownership, carrying or transporting of firearms, ammunition, or components or combination thereof, other than those expressly authorized by statute, is invalid.

UVA is a governmental entity (administered locally BTW) here

UVA as an agency here.

Under the Administrative Process Act, an “Agency”
is defined as “any authority, instrumentality, officer, board or
other unit of the state government empowered . . . to make
regulations or decide cases.” Code § 2.2-4001 (emphasis added).
To this effect, Code § 23-14 states that all state-affiliated
four-year universities are “governmental instrumentalities;”
Code § 23-9.2:3 supplies the “governing body of every
educational institution” with the power to promulgate certain
necessary “rules and regulations.” Perhaps most significantly,
Title 8 of the Virginia Administrative Code actually denominates
University of Virginia which administers UVA-Wise within its text as the Commonwealth’s
“Agency No. 85.” 8 V.A.C. 85, Agency Introduction.

UVA as a corporate authority here (shaky, but do-able argument)

UVA Wise's Campus Police is a Department here

ALTERNATIVE ARGUMENTS:

UVA was created as a public corporation (page 7 of 116) and is subject to the Dillon Rule

If the Heller Case establishes that the 2nd Amendment is an individual right, then Breach of Contract where the Code of Conduct states:

"The College is a community of scholars in which the ideals of freedom of inquiry, freedom of thought, freedom of expression, and freedom of the individual are sustained. It is committed to preserving the exercise of any right guaranteed to individuals by the Constitution."