Your best path is use of the audio shield, stereo line inputs. You'll definitely need preamps to get the mic signals up to the line level the audio shield expects.
Specific guidance is difficult when we have so little info about the microphones and preamps. I looked at
that Amazon page, but the info is so limited. I couldn't find important tech specs, like the voltage gain or a schematic.
Many different types of microphones exist, with different analog properties. Virtually all with analog output give a tiny signal, so you will definitely need preamp hardware. How much the preamp needs to amplify the mic signal really depends on the mic sensitivity. Condenser type microphones (which need power to operate) usually give more voltage output than dynamic (just magnet and coil) types. Some mics are good at delivering the signal over a long wire if it's properly shielded, but others have higher impedance output which usually means you want to keep the wiring short. So even though that preamp board looks like it's designed for 2 signals, if your mics are physically far apart and they are the types that want short wires, you might need 2 separate preamp boards simply for the reason of sending the stronger signal over the distance for both to reach the audio shield.
To move forward, I'd recommend doing more testing with the mics and preamps. You will need a way to listen (with headphones) to a line level signal. You could program Teensy + Audio Shield with one of the passthrough examples. Before you start experimenting with the mics, first feel the line output signals from your PC into the audio shield line inputs. Make sure you can hear the sound the sound your PC sends on the headpones plugged into the Audio Shield. Or if you have a home stereo or similar audio gear with line level inputs and headphone outputs, you could use it for testing if more convenient. The key point is to get a test setup where you don't need to mess with the computer and loading new code simply to hear line level signals on headphones. The critically important step is to test it with known-good line level signals like the output of your PC or a stereo system or other audio gear with line level output.
Once you have a confirmed good way to listen, then start playing with the mics and preamps. Important to use headphones, so you don't get horriffic audio feedback from a speaker back into the mic. Placing the mic farther away and monitoring something that makes a distinctive sound can also help. If the mic is close and listening to you, it can be difficult to discern whether sound you hear really is from the headphones or just from the room. Over-ear headphones that physically attenuate room sounds can help for this sort of testing. You can also just tap the mic with your finder.
If the preamp is working, even if the gain is too low, you should be able to hear the faint sound. That Amazon page says "The on-board resistor R5 is adjusted by the magnification. If the magnification is not suitable, the resistance of R5 can be adjusted. It can be selected from 5k to 150k. The larger the resistance, the higher the magnification. If the plug-in resistor is used, the original chip resistor can be removed, and the plug-in resistor is soldered to the reserved pad on both ends of the original chip resistor."
If you get no sound in the headphones, that probably means the wiring isn't connected properly or the preamp just doesn't work. If you can't find the problem, your best path forward is to share photos of your wiring. Maybe we'll be able to see something you've missed. Better lighting and photos from multiple angles can really improve your odds of useful help.
If you get sound, but also unwanted noise, best to share photos and try to describe the noise. I could go on and on about common audio noise issues... but this is already long. Hopefully it at least helps you to move forward and get those mics + preamps working with the audio shield.
