I think it *might* be possible, but it would be a big hassle. If you included a large dishpan filled with loose, dry soil and put it in the sun, they might use that instead of digging their holes for dustbathing. I've had some success with this method, though it has to be a heavy pan, or else they'll tip it and make a huge mess.
Obviously, the larger the run, the less visible their destruction would be. But they won't do the job evenly, and like Pat said, they'll leave large rectangular "footprints" all over the yard. Your biggest problem would be, I think, that chickens just like to scratch up the soil. Though actually, if you covered the floor of the tractor with largish fencing wire, that might prevent the scratching and force them to graze more. But I've always thought a wire bottom on a tractor might hurt their feet because they'd try to scratch anyhow. So I guess I'd recommend against a wire bottom, unless you hear from other folks that their chickens tolerate that just fine.
You might be better off telling your neighbors they could use your tractor to start flower or veggie beds, and then you could just park it for a while. They'll do a great job at that! (I've got nine hens in my tractor right now, working their way across all 30 beds in my garden, fifteen feet a day. I probably won't even need to use the rototiller before planting, just rake it a little bit).