Yes, use an improvised signal probe from an interconnect cable and a coupling capacitor (0.1uf is fine, connect it to the center lead, probe with the other cap lead) plugged into a tape monitor input.
Then you will depress the tape monitor switch to cut it in (don't be surprised that the OTHER channel is quiet - this is using ONE channel like a scope for your ears) and probe the tuner points looking for the difference in volume levels.
Turning to the AWE-066 tuner board - there are several points where the signal level of one channel can be impaired.
here's a list, in signal flow order:
Q8 HA1196 chip pin 4 (L) and 5 (R) source
AWE-066 pin 16 (L) and pin 14(R) wiring before filter or coupling caps
Next is coupling caps C43(L) and C44(R) passing the signal to the Filter.
AWE-066 F5 - this filter can die in one or both channels. If you get to that point, and want help, post a picture of the top of the tuner board (the bottom is inaccessible) for visible probe points.
Then there is a muting circuit,
Has C57 in the tuner's muting circuit been replaced - hopefully it ISN'T a sky blue Sanyo CSSA type crappy capsistor!! That has caused problems before. This is a LOW probability cure.
The muting is Q10 and Q11 and they share the emitter(grounded) and base (muting control) connections.
Unfortunately they are hard to test in circuit, because it is hard to tell if the signal is getting hurt in the F5 filter or by Q11 in the right channel.
edit - something I forgot:
FIRST- Ohming the switch contacts directly is difficult as the switch board is buried, on the AWS-094 board, input is pins 42(L), 40(R) and output is pins 45(L), 44(R).
So take an ohms reading (with FM selected) from AWE-066 pin 11 (Left) to AWG-038 flat amp(rut roah!!) pin 12 to establish a baseline, then for the RIGHT channel AWE-066 pin 12 (Right) to AWG-038 flat amp pin 11.
Then, if the ohms test is good, you need to make the signal probe, so probe here FIRST:
AWE-066 pin 11 (L) and pin 12(R) - these are the outputs of the board, check them for equal levels FIRST.