I think it should be possible. Essentially you could construct a large L-shaped frame (each side being the same length) with two small flat surfaces at the ends of the L, and a source of a short sound -- a cap gun for instance -- and a microphone at the corner. Then you fire the gun and listen for the echoes, timing their arrival. How big the apparatus needs to be depends on how short your impulses are and how accurately you can measure: arms 10m long should be fine.
Now you put the whole thing on wheels, and find a very large flat surface (salt flat?) and, on a flat calm day, you drive it along in various directions, and do the experiment repeatedly.
There will be two problems to overcome:
- there will be reflections from the ground, wheels and so on, but if you dimension things properly you can arrange life so that these arrive at times comfortably different (earlier) from the ones you care about;
- there will be some dragging of the air by the structure itself, which is unavoidable but which you can work to minimise by making it have as small an effective cross-section as possible and by streamlining suitable bits of it (you may need to do wind-tunnel tests).
What you will find is two things (remember this is all being done on a flat calm day):
- by rotating the device and repeating the experiment at lots of angles but while stationary you will find that sound travels at a speed which is not direction-dependent, or that its speed is isotropic;
- by moving the device along with it turned at various angles to the direction of motion you will find that sound moves at a speed which is constant relative to the air, not the device, or in other words that there is 'aether' for sound, and it is, in this case, air.
What you will not get is a null result, as the Michelson-Morley experiment.