Stick to a switch and use shielded cable, relays put coils right next to signal paths, so any interference from your 12V lines would be even worse through a relay... I'm not sure why you're worried about it. Your weakest link is going to be in the solder joint to the copper on the main board, that's going to be where any crappy sound is coming from.
Also, you need to make sure your signal levels and impedance matching is correct. Impedance is less of a problem at audio frequencies but your signal levels are critical. If the traces you're talking about are before the tape pre-amp (IE. right off the tape head) then you're probably going to get noise issues no matter what you do, and your signal will need to be really, really tiny and protected from overages.
If the traces are post-amp then you shouldn't have too many problems and they should be somewhat close to line level.. The problem is that it's difficult to tell before cutting the traces and feeding in a signal if it's pre- or post-amp..