You may be able to if you pay full retail price for the phone. You won't get the discounted price because minors cannot legally enter into contracts. The credit check requirement may also pose a problem.
You can try and get a Sprint phone and have it flashed for a prepaid carrier such as MetroPCS, or anyone that doesn't require a contract or credit check, but you wouldn't be able to access Sprint's network, and they have a 4G network now.