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