This FAQ run by the University of Virginia explains who may borrow materials from a UVA branch library: https://www.library.virginia.edu/policies/circulation/#borrow-who  

Among those listed (who are not U.Va. students, faculty, staff or alumni) are: ​

• Individuals associated with U.Va. agencies by specific arrangement (such as employees of VTRC, VDOT's research arm)
• Virginia residents (at least 16 years of age)
• Faculty and students at other Virginia higher education institutions
• Independent researchers
• Faculty from OCLC Reciprocal Faculty Borrowing Program institutions

To obtain borrowing privileges, the patron is required to present current, valid identification at any Library circulation desk. A map of the university's library locations can be found at: https://www.library.virginia.edu/libraries/  Valid forms of identification described on the FAQ include:

• Virginia driver’s license or ID card
• Proof of graduation from U.Va. and an ID card
• ID card from a Virginia higher education institution
• Approved independent researcher application (available at circulation desks) and an ID card
• OCLC reciprocal borrowing identification card and an ID card

Questions can be directed to the University of Virginia Library: library@virginia.edu or by calling 434-924-3021.