I think a lot depends on the space. I recently did two such things using speakers for playback. Same basic idea - a location overdub onto an existing track with a lot of singers.
one with a large group in a church and one with school kids in a classroom. The school kids was in a deadish room and the bleed was actually enhancing to the sound. Not bad at all.
the church thing OTOH, was quite problematic. Even with careful attention to placing the speakers in the nulls of the mics etc, there were enormous amounts of spill off the back wall. But turn it down and then it was too low for them to stay locked in. I ended up dropping their cue mix down to only guitars and let that be the guitar reverb for the song! .
I wanted to do what eoats suggested- distribute my limited amount (12) headphones to the critical members - the 'musical leaders' or 'first chairs', as I have seen that work well in the past. They can "pull" the others along. but the choir director nixed that idea because he felt it would create "bad feelings" with the ones who did not get the headsets!
A trick I read about, but have never tried involves using one of those small FM transmitters
and asking everyone bring a portable radio with headphones. You need something with some decent range, not the kind you use in the car to go 2 feet from your iPod into the FM radio. You would want to test it out in the location well ahead of time. This probably worked better back in the 80's when everyone owned a Walkman and most of those had the FM radio built in. But if you hit the right garage sales, you might be able to collect quite a few of these cheaply.
Maybe you could stream it over the internet and everyone could log on with their Smart phones? Probably would not be in sync, but that would be funny.