If you're inclined to drive to VA Beach (about 1 hr away) Naval Air Station Oceana has skeet, trap, wobble trap and 5 Stand and a good amount of open hours to the public. I have shot there a bunch with my dad and have enjoyed it every time.
There's the Richmond Gun Club in Charles City but I never shot there, know they have registered skeet shoots. May be a members club??
Our small club is in Middlesex County, roughly twenty five minutes east of West Point, Virginia. The club is open on Thursdays and Saturdays. Skeet, trap, and five stand clay sports as well as pistol & rifle lanes. It's a members club but you can shoot as a guest. If interested, let me know.