Couple questions.
First is if the machine is running and the power is cut to it, will it still have the inertia to keep moving and entrap someone dumb enough to open the gates and reach in at the wrong moment? Reason I ask is if that it the case then you need to do one of two things. Either build a braking system to stop the motion of the machine OR you will need to secure the gates when the unit is cycling.
Second, is the entire machine running on 110 or is that just the control voltage?
Getting the gates to have a interlock switch is fairly simple.
As long as the motor creating the dangerous motion is controlled by a contactor, and the machine will immediately come to a full stop if that contactor opens and the inertia situation is not a problem, then simply wire an industrial roller switch to the contactor coil and mount it in such a way that it will remove the power from the contactor any time that the gate is open.
But again, this is ONLY safe if there is no inertia in the machine that will allow the dangerous moving parts to continue moving after the power is cut.
Now, how to address it if there is inertia.
pneumatic cylinders.
Assuming that this machine will somehow already require shop air, this is the thing to go with.
You can either use a cylinder with an air valve controlled by the motor contactor power and a gate position switch to hold the door closed (but not to CLOSE the door which would create a new hazard) or you can have a gate switch trip a valve controlling a cylinder to both cut power and you apply a wood block brake to the moving part stopping it's motion. Of course the brake thing can break stuff if it's able to stop things immediately (which it would have to for safety reasons) and there is enough inertia to carry the machine through a cycle after it's started and you are trying to stop that inertia with the brake system.
So the option that the gate has a cylinder attached to it that is pressurized when the cycle starts that is also configured to not allow the cycle to start without the gate being closed to begin with is the best option.
To do this you would need a cylinder that would create enough force that it can't be overcome by hand plus a safety margin, that was controlled by a valve the would fully release the pressure and vent the output port to open air when open so the gate could be slid open against the cylinder when it was not engaged.
Pictures of this build project would be great at this point because I am only guessing about how this thing looks and that my idea will work.