Seismic refraction is your best bet ($$$$). However, if the cavern system is present at a shallow depth, ground penetrating radar will also work (less $$$$). Also, are there any surface features that indicate the presence of the cavern? Such as sink holes, karst topo, etc.?

I have a friend who has been locating abondoned mine shafts using aerial thermography. He flys over the property looking for cooler air coming from the ground in realation to surface temperature. He has had fairly good luck doing this.

The appropriate new technology to check underground anomalies is generally a GPR (ground penetrating radar). However, you can always probe the ground using percussion drill hole. However, the site must have access for a percussion drill (or even a portable drill rig, if available). This is the surest way to check if the site has caverns.... by checking the penetration rate of the percussion drill rod. mark the rod every 6 in. and time it as it went down. The driller also definitely can "feel" where the cavities occur. By drilling some holdes, you can "map" the location of the cavern.
Hope that this help.

You have received some advice on what might work. I'd suggest you get some professional help by talking to a local geotechnical engineer. Give them the details and let them figure out what is the best way to approach it. You could spend a fair amount of money trying an inappropriate technique - but then you wouldn't know it was inappropriate, would you?? After all, it was recommended here and it has worked elsewhere!! Geophysical methods are notoriously cranky - finding out how to approach the problem from someone who has local experience is always the best first step. Who knows - your problem may even be well known, and perhaps even already mapped.