(An internal vacuum has a male component that completes a circuit within the faceplate, so when you plug in the vacuum hose, the system turns on. There is no switch.)

I'm not sure which wires go with which, so here's what I have:

The faceplate has one Blue and one White wire. The wires in the wall are: Red (taped), White (capped) and Black (exposed).

I think that I should attach the white to white, the red to blue. But what to do with the Black? Is it grounding? If so, what should it touch/not touch?

Thanks for your help!

The wires in the faceplate are nothing more than a switch.

The installer of the cable used 3 wire when only two wires was needed, so one is a spare.

This should be low voltage system, probably only 24 volts.

You can as a test connect any two of the three wires in the cable. Once you find the two that start the vacuum, connect those two wires to the face plate, any color to any color. Cap the remaining wire left in the cable.

Another way is to use a voltmeter and you should have say 24 VAC between the two wires that you need to use. When you use another outlet, the vacuum will turn on and the voltage will drop to zero.

Open another faceplate and se how it was wired or see if you can figure anything out from the main vacuum unit.