By putting the modeling selector in the middle of its range and connecting a 10k trimmer in parallel I was able to select the amp model that I wanted, and could do the same with tone and gain pots. My question is with the two 10k pots in parallel like that wouldn't the resistance be roughly 5k and will the controller agree with that resistance? Seems like I read somewhere that they are pretty particular about the impedence they see...
...Was thinking of using DPDT relays to do the switching
If you are using relays (and presumably a transistor or MOSFET to drive the relay), the Arduino will be isolated from the pots. A relay is probably the "safest" way to go, because the relay contacts can handle AC signals, as well as DC, and you shouldn't have to worry about what the pots are doing or if there is audio or DC control signals going through them.
Adding trim pots in parallel with the existing pots is unpredictable. Apparently it's working for you, but depending on the circuit you might risk damaging the Vox, especially if you turn the pots in opposite directions. So... I'd suggest you use the relay to switch-out the existing pot while switching-in the trim-pot. A DPDT relay can do this if one end of the pot is grounded, be4cause you can leave teh ground connected on both pots and and you only need to switch the other two terminals. The danger with this design is that there will probably be a "glitch" when the relay switches. And, you might need an additional relay to mute the sound during switching.
If you get a couple of manual DPDT switches, you can experiment manually before you build & program the controller circuit.