My thought would be that there would be a small surface mount box inside of the base cabinet itself and the air actuator would plug into that and in turn the disposal into the actuator. I am running new MC 12/2 wire w/ separate green ground. What would be the proper way to have this pass through the wall, as well as through the base cabinet?
The most common installation is a metal box mounted in the base cabinet. The receptacle is typically a "half hot." One outlet is switched for the disposal (pressure in your case), and the other is always hot for the dishwasher.
Surface mounting in the base cabinet is just fine. Just be sure to bond the box with the green screw or clip to your equipment ground.