Can you race? No, probably not. Most sanctioning bodies require a license and specific preparation to your car to meet the rules of which ever class you intend to run with.
That said there are many opportunities for performance driving (not racing) on road courses such as VIR. There you will receive professional instruction, in a controlled environment. If you so choose, you can prepare for and receive the aforementioned competition license through these programs. Once you have that and a car you can move on to actual wheel to wheel racing.
NASA is probably the most prolific in the Virginia area. Many board members and thousands regionally, including my self have participated in NASA HPDE events. You can get more information from their web site
www.nasaracing.net. Look under HPDE (High Performance Driver Education) that will have all the info you need as well as where and how to sign up.