That's a screwy design...
I don't see any DC bias/reference. With a single supply, the input needs to be biased at half the supply voltage. I'm guessing you have constant 0V or 5VDC at the amplifier outputs, before the capacitor and that's "killing" your signal. (The input & output capacitors block/isolate the bias from the audio circuitry.)
I've never used a Norton amplifier. I'm sure it can be used in a regular voltage-amplification application but I don't know the difference.
Normally, mixers are built-around a summing amplifier, which only requires one op-amp There's usually a volume pot on each input and microphone mixers have a separate amplifier stage for each mic. Again, if you have a single supply the +input on the op-amp goes to a 2:1 voltage divider.
Could 5V be too low for this?
You can get a rail-to-rail op-amp which goes from (nearly) 0V to (nearly) 5V (with a single 5V power supply). I don't know how close the LM3900 goes to the power supply rails. A regular op-amp will probably work but you might get clipping (distortion) depending on the actual signal levels.
USB power (from a computer) tends to be noisy but it might be OK for line-level signals.
...Depending on the application you might be able to use a passive (resistive) mixer. The circuit you have is actually a 4-channel passive mixer, which is the four 470 Ohm resistors. If you want to mix 2 line-level signals, you could use two 4.7K resistors, and that's it! No other components needed! That's actually an "averaging" circuit so if there is only one signal it will be cut in half (assuming 2 mixed channels) and there can be some "impedance related " issues but in many cases it works fine and there is no noise or distortion.