I know this has been "done" before...
..but I thought I'd revisit the idea. The UK first had a system somewhat like this on an aircraft cannon and then Honeywell model 40 Mk 18 grenade launchers do it too.
Ok, ammunition is hopper fed by a sorter... that's another mechanism altogether. The breech of the mechanism is split, and each half is formed into a rotating wheel, one clockwise one counterclockwise. When they turn together they "trap" a live round in the breech. There are an even number of chambers, so as to have a chamber "notch" directly opposite the live chamber in the wheels at any given time.
The bolt mechanism is an interesting quandry... it's likely a heavy forged aluminum piece. The firing pin mechanism is captive within and releases the pin when the bolt is settled against the back of the breech. There are also two "fingers" on either side that fit into the chamber notches opposite from the live chamber in the wheels, which serves to lock the rotation temporarily so the ratchet mechanism sees no stress and holds the two chambers together to not stress the bearings out. When the round in the chamber is fired a gas piston is operated that kicks the bolt back, releasing the breech wheels to rotate, and when the bolt reaches it's rearward limit it strikes a ratchet mechanism that rotates the breech wheels which form the next live, loaded chamber. If the trigger is released at that point then the bolt is locked back. To unload the weapon you'd leave the bolt back and lock it, interrupt the flow of ammunition to the wheels, and run the ratchet mechanism by hand (likely by lever) several times to roll the remaining ammunition out of the breech wheels.
Yes, sounds complicated. But, because of the way the mechanism works (gas operated straight blowback bolt with active firing pin) you essentially have a firing rate variable from semiauto all the way up to as fast as you can move the bolt with gas action... or as fast as a cartridge can fall half an inch into the chamber.
The *unfortunate* part is that the mechanism will require a straight-walled cartridge with no rim.... something custom, but at least not difficult. The cartridge does not require an extraction rim, as extraction is accomplished by simply opening the chamber and letting the rotating breech fling the cartridge out directly downward. The UK version used an aluminum cartridge with those qualities and an enclosed bullet, i.e. the case mouth was even with or beyond the nose of the projectile.
Depending on the width of the chamber walls, you could in theory even make the cartridges out of plastic, depending on the rate of fire and heat absorption and dissipation.
Mind's running wild again, and I like it :) I *need* a copy of autocad....