Can a person with a hit and run felony buy and own a handgun in the state of Virginia?

No. Federal law prohibits any person with a state or federal felony conviction from possessing a firearm. The penalties are steep. Only if your rights have been restored under both state and federal law, may you possess a firearm. In Virginia, the ban applies unless: "You have been pardoned or have had your political disabilities removed pursuant to Article V, Section 12 of the Constitution of Virginia, and if the executive order does not place any conditions upon the reinstatement of your right to ship, transport, possess or receive firearms." or "You have been granted permission by the Circuit Court of the jurisdiction in which you reside to possess or carry a firearm (with no restrictions on the type of firearm) and one of the following: you have had all other political rights restored by the Governor, or, you have had your federal disabilities removed by the Bureau of Alcohol, Tobacco and Firearms" or "You have had your political rights restored by the Governor of Virginia or the Governor of the State in which you were convicted and one of the following: the reinstatement of rights included the right to ship, transport, possess or receive firearms, you have had your firearms rights restored by the Virginia Circuit Court in the jurisdiction in which you reside, or you have had your federal disabilities removed by the Bureau of Alcohol, Tobacco and Firearms." Every state has different rules about whether and how you can have your rights restored. Your best bet is to check with a criminal defense lawyer in your jurisdiction to see if you are eligible for the restoration of your right to possess a firearm.