Can someone help me understand the law concerning under the age of 18 and handguns?

Per the NRA's site it appears both my son and I could be in trouble if he were to carry a hand gun. He is under 18.

Yet they also indicate that target shooting or hunting are exceptions. So if he is out on his quad with a 22 pistol on his belt, hunting and target shooting is he ok?

You need to check your state laws very carefully. Federal law prohibits the possession of handgun by anyone under 18 (unless they are in the performance of military duties). There are some exceptions for hunting and targeting shooting, but the juvenile is usually required to be in the company of a adult who can legally possess the handgun. There may also be a requirement that the land be owned by either the juvenile, his family or his employer.