How do I multiplex many signals into my sound card?
Let me be completely honest with you.  Don't do this.  I would compare this approach (combining 24-channels of piezo info into an audio input) to trying to move all of your furniture from one house to another with a sports car.  You can do it, but you are much better off just renting a moving truck.

The typical way that you'd do this is to get several ADCs.  Not ADCs made for audio.  But something with a sample rate of 200+ KHz and a mux on the input.  Most of these muxes will be an 8 or 12 input mux, so you will need several ADCs to get all 24 input channels.   

Then connect these ADCs to a microcontroller (MCU).  Some MCUs already have a Mux+ADC integrated in them, but I doubt that you'll find one that can take 24 channels (but I could be wrong).  I would pick something like an ARM Cortex-M3 or M4 from T.I..  Some of them have USB interfaces, which you can use to connect up to a PC.

This approach sounds like a lot of work, and it is!  But the amount of work is less than what you were thinking, and the odds of failing are lower.  

Good luck!