Q: " Would something like this be legal without a permit?"
A: No.
With the application of motor power to a Gatling gun, it becomes a machine gun. Just closing the switch or engaging the clutch to apply power constitutes "a single operation of the firing device."
For a minute there, I thought you'd had the same idea that crossed my mind many years ago: Turning the barrels by human power through the operation of pedals. It never fascinated me enough to explore the legalitiy of such. I was more charmed at the idea of a half-sized reproduction of the classic Gatling Police Gun, complete with both artillery carriage and tripod mount.
THAT, we know is legal.
Unfortunately, it is also quite expensive by the time you pay someone to do the intensive machine work required.
Best,
Johnny