That's just two inverting stages with gains of 2.4x and ~9x, respectively. Gain is the ratio of the two resistors on IN- as usual, and IN+ is just some virtual ground. The first stage automatically biases the DC level of the signal to the virtual ground.
So I assume output stage that attaches to the STAC9708 is the higher-gain stage, correct? Also, I don't really understand what you mean by "The first stage automatically biases the DC level of the signal to the virtual ground.".
Reducing 10kΩ OTOH will increase loading on the mic and reduce low frequencies.
What about changing either the 24k or 18k resistors on the OP-AMP? That way there will be no change to the mic load?
It's probably easier to build a replacement from ground up on a perfboard and leave the poor card alone.
While I agree, I'm doing this to sort of challenge myself to repurpose old hardware (this is an old PCI card which I likely won't ever plug back into a PC, but if I can still use it and reduce ewaste, fantastic) and learn a bit at the same time.
Drawing both of the schematics was a fantastic learning experience for me, and being able to tinker with (the correct) resistors and understand how they affect the op-amp will also be a further good experience.