IIRC, they hooked it up to an electric motor. What they created WAS a machinegun. They are allowed to do so because they hold the proper licenses.
Like others said though, so long as you were activating each shot by some sort of mechanism, it would not be a machine gun. (Think old school hand-cranked gatling guns). If you put a motor on that gun and push a button to activate, it's a machine gun.
You're correct on the licensing. They hold FFL07/SOT02 status that allows them to manufacture and register dealer sample machine guns.
They wouldn't be allowed to manufacture a motorized Gatling gun and then sell it to an individual though, unless that individual was also a dealer with a law letter or was a requisitioner for a law agency.
Sons of Guns has potential to be so much more than it is. I'm not going to go over the shortcomings that has been discussed enough. I hope the new season see's some improvement.
I will admit that the new show, American Guns or whatever has actually made me miss SoG! If it weren't for the hot (and unfortunately underaged) daughter, the new show would be totally without merit.