Check out www.bringfido.com. It has a lot of information about traveling with your dog, including pet-friendly hotels, hiking trails, etc. You can check by state to see what's in VA. However, all of the ones we have gone to (in MD) all have had leash laws. Not sure if there are any parks where you can take a dog off-leash on a hiking trail.
We are looking to visit some national parks this summer with our dog, and many of them don't allow dogs at all. Some parks allow dogs only on certain trails. All of that information can be found on the National Park Service website. State parks have similar information on their websites as well.
Good luck!