I will be buying the 3-Axis ProboStep Stepper Motor Driver Kit. Is it possible to have the kit control the trim router using a relay board?

You'll need to use their relay board, and probably the breakout board.

I looked at the wiring diagram for the relay board and I'm not sure about connecting the router motor to the relay board. The breakout board comes with the kit. The relays are rated for 12 Amps at 125 VAC. The Dewalt trim router uses 5.6 Amps when running but what about the power needed to start the router running?

You can use the driver board relay to control a heavier external relay. Just find one with a 120 volt coil.

John