This is timely advice! We were in a movie on Dec 26th at Downtown Disney, due to fly home, when I (had my phone on vibrate and on my lap) got a message from Southwest that our flight was cancelled. I stepped out, and got on the phone with Marriott and got connecting rooms at the Rennaissance at the airport (our car was packed with our luggage, since we'd already checked out of the Boardwalk Villas). I had no luck with Southwest, so thought I'd just wait till later to call, and the website wasn't helping me (I think they were swamped and the wait time on the phone was 2 hours).
Turned out, I booked my husband the next day on Airtran in business class. The kids and I couldn't get home for until the 29th, so I just kept the rental car and we enjoyed the hotel (it was very nice!) and we even spent a day back at the MK and Epcot (only $5 extra to add a day to our tickets).
Luckily the kids were still on Christmas break, and we could afford to spend the extra on the hotel and meals (but if we'd had
travel insurance it would have helped with that, so I'd definitely buy travel insurance next time).
I heard there were tons of lines at the airport and people sleeping there even. I do know the Hyatt was booked up, since I tried them (thinking it would just be one night). Glad though - my husband is Marriott Platinum so we got club rooms with breakfast and snacks included.
If your flight is cancelled, the last thing to do is to go to the airport! Book a room and then call or go online.