I've built a Valvecaster for someone but now they've asked if I can make it run off 18v.
What voltage is it running at now ?
If I throw a LM7812 voltage reg in there, is it likely to introduce noise into the circuit?
No, but you may want to use the suggested output filter component[s.
Or would I be better off just getting a large ceramic resistor (of appropriate value) to drag the voltage down?
If you're dropping from 18v to 12v using a 7812, it might get warm/hot, I've never made that V-drop/current with one though, I'm sure others have. the V-reg has the advantage of providing a smooth regulated voltage. Another option would be a zener diode.
The V-reg is pretty easy to work with, make sure the chip drawing has the pins labeled right under it, even if you have to de-code and redraw it off the data sheet.