Just thought I'd share a bit of our experience during Hurricane Frances in 2004. We were scheduled to arrive (driving down) on Sunday the fifth of September and were really torn whether to cancel or not. Like many of you, I recall watching the radars and cones, looking for any small wobble one way or another. At its peak, Frances was a Cat 4, and we agreed that we definitely weren't going to risk that. Fortunately, she weakened to a Cat 3, then was predicted to make land as a Cat 2. We were staying at Shades of Green that year and called the resort who told us to come on down, if we wanted to make the trip. So, we decided to go but ended up having to hang north an extra day as Frances slowed down (actually did a little wobble backwards!). All of the Disney parks were closed September 4 and 5, with the Magic Kingdom, Epcot and Typhoon Lagoon opening Monday the 6th and all reopening Tuesday (I believe). I will say that it was just my husband and myself, and our decision to travel might have been different if we had kiddos, as we did experience some nasty weather on the way down. The hotels on the trip down were filled with folks evacuating north which was quite the experience.
Again, we were not on property during the hurricane but arrived immediately following, and Disney does do a great job getting things going again. But, keep in mind that it's not only the physical effects on the parks, etc., but the cast members as well. On our trip, we heard of cast member staffing issues, as they had damage at their homes to address. Even if Disney isn't in direct line for a hit and the parks are open, depending on the intensity and size of the storm, there still could be issues with events limited or cancelled. I do recall reading live postings of folks here on the forums who were in the parks during other storms, and although Disney does its best, being holed up in the resort isn't really all that magical (long lines for less than stellar food at the food courts, etc.).
I will say that after things settled, we had gorgeous weather with lower crowds as many folks did cancel. And, this actually ended up being one of our favorite adventures (and adventure it was!). But, please listen to the experts and don't put your family at risk, if things look bad. We all know how much time, energy, money and emotion are invested in a Disney trip. But, no vacation is worth risking life or limb. Good luck to everyone and please stay safe!