Hi, Rob,
I'm not as familiar with DE games as with Williams, but then the DE hardware seems to be similar to Williams System 11. From your mention of there being 4 switches, I take it that these are the switches poking through the playfield, which are closed by a ball hitting the rubber band. Is there another switch for each slingshot, operated by the solenoid linkage(s)? If so, then the paired switches are used to control the solenoid transistors independently of the game program and Cpu; a disadvantage of this is that if a switch remains closed, the solenoid remains energized, quickly burning out the coil, transistor and traces on the circuit board. Williams used to use this on Jet bumpers and slingshots, calling them 'Special' switches/solenoids. I think DE used the term 'Direct'.
The capacitor may well be the remains of an 'RC' (Resistor-Capacitor) network, used to improve the sensitivity of the switches so that the solenoid gives a good 'kick' without needing to have the contacts too close together. With the contacts open, the capacitor charges up via a 100 ohm resistor (missing in your case?). When the contacts close, the capacitor discharges, prolonging the input pulse applied to the solenoid transistor.
Note that Special/Direct switches aren't part of the machines 'switch matrix' - each is simply an input to a logic circuit, controlling one of the solenoid transistors. As such, there's not a diode used on these switches. The program awards scoring, generates a sound, alternates other lit features, builds up a bonus (or whatever other result is needed) by detecting the switch operated by the linkage.
As to the wiring, if there's nothing in the DE paperwork, a tip is to refer to the manual and/or small 'handbook' of an earlier Williams game, e.g FirePower. The resistor and capacitor are connected in series (using a 'dead' lug in the switch block) across the switch contacts, with the '+' lead of the capacitor towards the 'input' side of the switch, i.e. the wire with a unique colouring - the other wire is a ground.