In Southern VA you have Blue Ridge region SCCA which runs quite a bit in Verona which is not suitable for karts due to the severity of bumps. They do run Salem civic center and Roanoke civic centers which are a bit on the gravely side but can be done in a kart.
If you near the coast then TideWater Sports car club. They have the ACU-4 pad which is a nice site.
If you want to travel a bit to NC, you have Triad Sports car club and Central Carolina region SCCA. Both are very welcome to karts and have some of the best drivers in the area. Also if you have not done an event at Zmax drag way I would highly recommend it.
Triad has an event this weekend in Winston-Salem NC. auto-x.com
Tarheal Sports car has NCCAR in Roanoke rapids but most of the time the course is just too fast and not enough autocross for a kart. Would be great fun in a car but very boring in a kart that can stay flat though all the gates.
Later this year there may be some events at Danville airport. We are currently on hiatus there due to the runways being repaved. They say it should be done soon, but we are not sure of the currentcondition of the lot.