The idea is to measure continuously while the microphone moves along a planned route within the 2D plane. Humans would be around and ideally, it should stop if it runs into any resistance. It doesn't need to carry much load/weight (just the microphone and cable) and it requires little force.
More speed typically means more noise - so I'm not picky about speed. Precision - well anything within .1mm should be good enough. The initial use would be indoors. If things work out, it could be nice to take it somewhere even outdoors - but I suspect portability introduces a whole new set of complexities.
I took TygerDawg recommendation and looked up the Cartesian style robot. I think this would work best. ZX robot is what I envision with linear motors. A vertical arm (index up or down) with a X arm to move in and out. Then there is the software to run it (hopefully something simple?)
I just don't know where to find something like this that is cost affordable.