First I would add a 5th microphone, so that the maths are simpler. (edit: what are the minimum number of micro/hydro-phones needed to perform 3d tracking?)
I would measure (as precise as you can; e.g. with a laser) the separation between each pair of microphones and the distance from multiple reference points.
Then we need a little bit of maths to solve the multiple measurements together with the dime delays into a source location.
Traditionally, if the locations are known as x,y,z then maths is straight forward and discussed also in this SE. The interesting aspect is to use measured microphone distances to estimate also microphone locations. I have not done that yet but may try.
Edit: it all translates to triangular geometry
Edit2: here comes a basic outline of the procedure
