I think this could perhaps be done with some PCs (or any stereo digital recorder) recording audio. Two might do it but three would be better.
You need two microphones wired up to the stereo input of the PC. Place the microphones carefully at surveyed locations that you can put on a map.
Record the audio.
Using Audacity Software which is free (I have NO experience of this software) you will I think be able to obtain the difference in arrival times of the sounds on the two channels. From the speed of sound (record the temperature too) you will be able to get a position line from the pair of microphones.
Use a second PC and microphones at a separate location and you will get two position lines and a position.
With three at the approximate corners of a sufficiently large equilateral triangle you would get a very good position in all cases.
The microphones would not need to be too far apart but I would put them a few feet apart or as much as I could easily manage. The digitised sound has a sample rate of maybe more than 50kHz. I think that this will provide a VERY accurate position line if the microphones are even a few feet apart.
Note:- No absolute times are necessary with this method other than to ensure that you have the right gunshots on the separate PCs.
I have NO idea if this will actually work but you could get the software and try it with any old microphone to see if you can actually identify the gunshots on the Audacity level graph and how precisely they can be identified. There will I am sure be many echos and other confounders.
You also need to check if you can zoom in sufficiently with audacity to measure time differences of small fractions of a millisecond - my guess is you can
The further apart the microphones the better the position line.
This may not work at all, I don't know but it is pretty cheap to try out.
If you proceed someone here may be able to do the math (maybe me

for the position line if that is not your forte.
I would synchronise the time on the PCs of course to make it easy to identify the particular noises you want to look at.
Very likely someone will be along soon to shoot me down;-)
Good luck.