None that are near, outdoors, free, and allow handguns, but you get some of those.
Free, outside and "legal to shoot there" limits your choices and forces you to drive a long way. Amelia WMA is free and outdoors, is to your east about 1.5 hours drive towards Richmond,but prohibits pistol shooting for no known reason.
Northwest of you about 2 hours, near Augusta Springs,VA is Hite Hollow's outdoor range, managed by the USFS. You can shoot anything there,but you literally have to drive over mountains to get to it from your location.
Rivanna Rifle and Pistol (membership required) is about 1 hour 20 minutes north of the park. There are at least two paid ranges in the Lynchburg area: Izaac Walton League and Lynchburg Arms (indoors and paid). Both of those are inside of an hour's drive.