A simple DIY method would be to use 2 contact microphones.
"Contact microphones usually are piezo electric discs with a weight attached to them, and they are designed to pick up sound waves in solid materials. You can find them very cheaply on Aliexpress / Ebay / China / Etc, or buy higher quality ones in your local muzic store. They are commonly used to stick on insturments, to pick out the sound of that single instrument in an orchestra.
If you use 2 of them with a stereo input and some sound analyzing program, you can get a rough estimation of the location of the critter by the time delay. If you measure no phase difference, the critter is in the middle. If it's closer to one of the microphones, you get a phase difference.
Also, once you've managed to record the sound signature of the creature, you can probably find what animal it is from some online database. And this may help in narrowing down in what it's habits are, where to find it, extermination methods, etc.