Virginia Motorsports Park in Dinwiddie (drag & autocross)
Richmond Dragway in Richmond (drag)
Richmond International Raceway in Richmond (big roundy)
Langley Speedway in Hampton (small roundy & autocross)
Southside Speedway in Chesterfield (small roundy)
Summit Point Motorsports Park outside Winchester (three road courses)
Virginia International Raceway outside Danville (big multi-config road course)
You can drive a street car on any of those tracks - except RIR and Southside Speedway AFAIK - during various events each year.
Also worth mentioning are:
G-Force Karts in Richmond (indoor arrive & drive)
American Indoor Karting in Portsmouth (indoor arrive & drive)
VIR's Plantation Valley Kart Track outside Danville (outdoor arrive & drive)
Plenty of places to scratch that itch without involving public roads!
Jon