Look for EPSG:3968, NAD83 / Virginia Lambert. There are variants using different geographic coordinate reference systems (CRS) like 3969 on NAD83 HARN, 3970 on NAD83 NSRS2007, and 6591 on NAD83 (2011). The definition came from the Virginia Information Technologies Agency.
Because it uses the Lambert conformal conic projection, that means that shapes/angles are maintained, like a State Plane zone. Here are the projection parameters:
Central_Meridian: -79.5
Standard Parallel 1: 37.0
Standard Parallel 2: 39.5
Latitude Of Origin: 36.0
False Easting: 0.0
False Northing: 0.0