There is not an age restriction in Virginia law for registering, titling or insuring a vehicle in your name.
According to a representative of the Virginia Department of Motor Vehicles a minor can be an owner of a vehicle but when they register the vehicle they will need to show proof of having the minimum Virginia Liability insurance on the vehicle which may be an issue if you cannot find an insurer to write a policy in your name only when you are a teenager under the age of 18.
In VA and most states, it is up to an automobile insurance company whether they will write a policy for a minor. Generally a car insurance carrier will require those under the age of 18 (and thus still a minor) to have a parent or guardian sign with them on their insurance documents since they are considered legal contracts.
The VA DMV did not state that a parent would need to be on the title or registration of a minor however you may want to contact them directly to make sure you do not need a parent or guardian to sign with you on this paperwork if you plan to buy, register and title a car in your name when you are under age 18.
If you are buying a car from a dealership (or even a private party) it is likely that the seller will also require a parent or guardian to sign or co-sign with you because normally when you sign on contracts as a minor the paperwork may not be binding unless an adult of this nature signs with you.
Once you turn 18 years old then you should have no issues buying a vehicle or car insurance without the constraints of having a parent or guardian signing with you. At this age you will be considered an adult and your contracts would not be voidable as they would be when you were still a teenager under the age of 18.