Should I let my teen drive if he does not sleep much?

Probably not. If you are concerned as a parent about sleepiness and mental acuity or driving ability then as a parent you reserve that right. Sleepy driving kills. Also teens do not have experience to help them in situations to think more clearly in a sleep deprivation setting.
Probably not. One needs to pay attention to the road while driving, if he/she is sleep deprived the chance of getting involved in an accident increases, it is preferable to have a teen that angry than a teen that is gravely injured.