The injunction wouldn't prevent the district from operating... it would constrain their actions - most probably around the actions defined in the contracts in dispute. And considering the new board is already in place and has already transitioned, it's unlikely the court would force them to go back in time and put the old board back. At best you might get some sort of receivership/proxy leadership while it's in dispute. But I think even that is far-fetched. I would expect any injunction to scope just the board from messing with the existing land management plan and force them not to pass any new resolutions that would be considered targeting disney specifically.
The court isn't going to shutdown the government - it will find some sustaining operating point if it got involved at this point.