There are quite a few along I-66/Delaplane that I think are pretty decent and accessible to Ffx Cty.
Barrel Oak Winery is a good place for kids to run around, not snooty, kind of a Spring-Break-meets-35 atmosphere. Their wine isn't that great, but it's very scenic and fun. It's also dog-friendly and definitely family-friendly. They often have live music and some kind of a food truck on the weekends.
In the same area is Delaplane Cellars which I thought had good wine and also nice scenery. It's a quieter atmosphere and a bit "snootier" than BOW. They were still welcoming to my toddler, though, and he enjoyed seeing all the tractors and farm equipment out there for harvesting time (is that now or has it passed? I can't remember).
Three Fox Vineyards, same area, is also nice. Atmosphere is somewhere between BOW and Delaplane, but I didn't like their wine quite as much as I did the first time we went (we went most recently last fall, so it could be different this year). It's very, very scenic. They have tables set up among the vines for enjoying wine or snacks. There's also a creek at the edge of their property that's a great place for a picnic.
Aspen Dale is also cute, a horse farm, and does good pairings of wine and food.
There are a bunch of others in those exits around Delaplane, too, but I haven't tried them.
There are lots of other great wineries around VA that are scenic and have good wine, but I like this area because it's close by and the wineries are close together. Not sure what PP is saying about them being snooty and family-unfriendly, because I have always felt welcome and see lots of families there. Some do get crowded, though.