It will depend upon state laws at what age a person can legally have ownership of a vehicle. State laws can differ from one state to the next regarding the minimum age they will permit a person to own a motor vehicle.
In general though a juvenile cannot enter into a contract so thus cannot be a sole owner of a motor vehicle. A minor cannot normally actually own property. His or her parents technically own that property. If a parent will co-sign on a loan for a car, the juvenile can buy a car, but the parent will be the legal owner.
There can be exceptions to this generalization depending again upon state laws. For example Arizona has an emancipation statute for minors over 16 who can show they are sufficiently mature to handle their own financial and health-related affairs. Here a minor to petition the court for an order of emancipation. If the order is granted, the minor would then be able to enter into legally-binding contracts for the purchase of property, including cars.
Even those states that will permit a teenager to register a car will normally require a parent or guardian to sign on any legal documents or contracts since they are still a minor. The Ohio Bureau of Motor Vehicles (BMV) states if you are under 18 and want to title a vehicle, your parent or legal guardian must complete a minor consent form. A parent or legal guardian must also accompany you when you appear in front of the Clerk of Courts title office staff to have a vehicle titled in your name in Ohio if you are a minor.
To find out for certain what your state's laws are regarding the minimum age requirements for owning a vehicle check with your Department of Motor Vehicles or like state agency.