Can I add an audio jack to a digital audio board?
Having an oscilloscope would be ideal, but not obligatory.

If you have a multimeter, you probably can find if there's a ground in the potentiometer. Then you can test with a low power audio amplifier if you find useful audio signal between that ground and another terminal of the potentiometer. Keep a 1uF capacitor (non-electrolytic) and a 1 kOhm resistor in series with the test wire. If you find audio, measure with a multimeter, is there DC, too. 

If there's DC, it must be separated with a capacitor. The needed capacitance depends on the allowed bass attenuation and the impedance of the connected audio input. A 1kOhm series resistor prevents a short circuit which is possible when connecting an audio plug.

Seems to be a stereo unit => you need 2 resistors and capacitors. If you must use electrolytics for high enough capacitance, they must be connected with right polarity to prevent the wrong polarity DC to break their insulation.

The mechanical construction also is a challenge due limited space and generally in equipment which are transported there's an incredible amount of thumps and vibration. They break connections which can repeatedly get bended forth and back.