Yes, I don't see why it shouldn't be doable. Mach3 can run 6 axis of coordinated motion and control the I/O's for the solenoids etc. If two LPT-ports doesn't give you enough I/O then you expand with a MODBUS enabled PLC etc.
The discrete outputs (and inputs) can easly be controlled by M-codes that you call from your G-code program or you can write an M-code that does the whole sheet loading cycle.
Or you can program a PLC to do the sheetloading/unloading and just signal it from Mach - lots of different ways.... ;-)
Cool animation by the way!