I don't know of any reputable chain that rents to under-21s, and I would think that the types of places that do rent to that age group would most likely not have the support system in place that you would need if something might go wrong with the car. These would be smaller mom & pop places, which wouldn't allow rentals for that type of trip anyway.

Buying and selling a car is a pain, dealing with the licensing, taxes, registrations, insurance, etc.