Yes you should be able to do that. I didn't fully see what that switch was doing, but in general you can replace a switch with a relay. If you need to have a 3PDT relay you could certainly find one on Digikey, Mouser, etc. You could also use 3 SPDT relays, but you would want to make sure they all switch together.
Depending on the relay, you should be able to drive it with the motor shield or any other motor driver. You would connect one side of the coil to ground and the other side to an output of the motor driver shield. Now whenever that output goes high the relay will switch on. You can control the relay by commanding the corresponding 'motor' to move or stop.
If you are only driving the relays you could also just use a simple MOSFET to control the relay coil as an alternative to the motor shield. If you have multiple relays you could likely stack the coils and so long as you have the voltage to drive the 2-3 coils you could control them with one output and guarantee they switch together.