You need something to convert from parallel to serial and back again. On the receiving end, this is pretty simple for a bunch of relays - just some shift registers, basically. Put some little drive transistors on each output to drive the relays, good to go. There would be 3 wires - clock, data and a latch signal. Maybe a ground to avoid noise. Get the power from the car at each end.
On the transmitting side, it's a bit trickier. You need parallel to serial conversion, a clock, something to send the latch signal after the final pulse, and if that membrane switch is matrixed, you are going to need to read a switch matrix, if you are trying to sync up with an existing thing that's trying to read the matrix, it could be a real *****. Doable with a microcontroller, but it's a fair amount of circuitry for discreet logic. You would need parallel to serial shift registers, binary counters with presets, couple of 555's, not sure what else.