Mickey and Minnie Mouse Ponchos, just like at the parks. Organizers and spectators of other sports that compete outdoors are able to cope with rainfall and successfully produce multi-day events. Us cheer fans could stand to toughen up a little bit anyway:)
The stage would be covered, the warm-up areas would be indoors, the only people getting wet would be the spectators. The organizers could monitor the weather and have contingency plans in place. Actually, you could have multiple stages running. Two indoor stages, and one outdoor "super-stage" that features the most popular divisions to allow everyone to watch. Parents/familiy with kids competing on the super-stage get a VIP-like credential to move to preferred seating. This way you don't need to be running from 8:00 am until 10:00 pm on both stages. If you need to have a "rain delay" for 2 or 3 hours, no big deal, because you have downtime built in.
If it is hurricane-like conditions, then force it back inside and let everyone sit on each other's laps - or, more likely, make people stay outside because the fire marshall will be shutting it down as it continues to get bigger and bigger.
Just throwing it out there.