Corrected entry: No way Kersey leaves the gun shop same day with a full auto AR-15. It would take boatloads of paperwork, waiting, and registration with the federal government. He could have bought a conventional semi-auto AR-15 much more easily.
Cinderdan
8th Mar 2018
Correction: There is no such thing as a full auto AR-15 unless someone has changed the firing mechanism in the lower receiver. You can not buy a full auto AR-15.
Kersey didn't even know how to shoot a pistol and had to practice with the throw down gun. The gun shop sold him the AR-15 capable of full auto. He would not know how to convert it.
Cinderdan
You don't need to know how to shoot to do a full-auto conversion. There are kits available with instructions online. A bump stock could also be easily added to the firearm to make it functionally full-auto even if it technically is not.
BaconIsMyBFF
Bumpstocks are a terrible gimmick that make firing and aiming properly very difficult. They are nothing more than a toy.
The gun store couldn't sell him a full auto AR because they don't exist, as the OP stated. There are military versions, like the M4 carbine which can be had as NFA items, but not in Chicago.