So of course the dealership's system was wrong and they didn't have the part it said they had. After doing some poking around under the dash I can see three different switches. Two of them that are on the clutch pedal and one on the brake pedal.

The two on the clutch pedal are the same switch/part number, but it looks like one of them is always open, and one of them is always closed. One has a red w/black stripe on it and a brown wire going into the connector, and one has three brown wires coming into the connector. Does anyone know which of these two control the start sequence? I've got only one ordered, so when I get it tomorrow, I want to know which I need to replace in the hopes that I get the car working.

Edit: I just borrowed a multimeter and checked the switches. Both have no resistence when opened and do show a resistence when closed. That should mean they're both functioning properly? Could it be the connectors themselves going into the switches?