Firearms Owners Identification cardd.
Not legally in Illinois.
a) In certain municipalities he could if the type of gun violated that municipalities rules, and/or if it is not properly broken down and otherwise made inaccessible. For example, handguns are banned in Chicago and several other cities regardless of whether or not you have an FOID card. There is no concealed carry in Illinois, nor open carry that I'm aware of though this may vary from town to town.
b) always, unless (I think) you're not a resident of Illinois.
He couldn't buy it, nor possess it if not accompanied by an adult with a valid FOID card.
But I'm not a lawyer so take what I say with a heavy grain of salt.
eta: I notice that there is no longer the requirement to have your FOID application notarized, it used to be so a few years ago. Also LOL at the "banned" tag from the forum! And now "tag" has been tagged!