Where in VA? I'm more south east. There are matches at VIR (Virginia International raceway at the NC border), and matches at the Roanoke rifle and Revolver club. Wrong direction for you though. Check out practiscore.com to search matches also.
I've lived around RVA a couple of times and comps (let alone outdoor ranges) are few and far between. I'm not sure what's down by VB and Chesapeake or far out west, but it's slim pickings within an hour of the city. I've never lived there long enough to join one of the club ranges (which is pretty much your only option), but the only thing that intrigued me was the open matches at Quantico. Although I never made it to one, I saw a lot about them on VA Gun Trader (I think) and the VA shooting reddit page. I heard Cavalier in Montpelier is the best range/IDPA, but it's a year wait list for membership and I never actually went.
Side note: If you're thinking about going to the public game lands on a weekend to shoot a few rounds or zero, I would highly suggest you don't. Early weekday mornings is your best bet or you're putting your life in some sketchy peoples hands. I had a hankering to shoot one weekend and it was so sketchy I feel obligated to mention it.
Sorry for the late reply, finally got settled in. I looked at PractiScore and didn’t find much other than VIR near me as I’m in Charlotte Co. Kinda disheartening moving from central Tx where a range is literally 10 minutes from my house.
VA has a lack of public ranges because most of us can just step out to any number of privately owned areas to conduct our training. Personally I despise public ranges. I've dealt with way too many ego trip RSO's, former "green beret Nam vets" that weren't born until 1972, and militia wannabe's at most of our local public ranges.