Hi there, Elaine!
In Florida, summertime thunderstorms are a daily occurrence. Walt Disney World will not close for "regular" thunder and lightning storms (their posted "flash/bang" ratio, however, is adhered to for outside rides, water Parks, etc.), and the good news is that not only are they reasonably brief, but the Parks tend to empty out. Grab a poncho and make the most of the reduced lines!
In the event of a hurricane, and you have booked your package through the Walt Disney Travel Company, you will have certain protections under
their hurricane policy. But the weather has to be pretty severe to close; during this most recent Tropical Storm (Debby), for example, the Parks have remained open to the public.
If the rain is so bad that it inhibits your time for the entire trip, you can address that with the Walt Disney World Travel company, and they'll handle your case on an individual basis.
Should you opt out of visiting the Parks, there are a variety of activities available at the Resorts for children and families. You can hop on the monorail and tour the Resorts on the monorail line. You can head over to Downtown Disney and do some shopping, or challenge the games at DisneyQuest.
I think you'll find, though, that with some packed ponchos, flip-flops or Crocs (avoid socks and sneakers on rainy days - wet feet make for an uncomfortable experience!), and your senses of humor, you'll be in for a fantastic visit - rain or shine!
See you later, celebrator,
-Deborah B.