It tends to be a relatively uncrowded time, September is peak hurricane season. As WDW is inland, it is generally not directly affected, though if there is a hurricane offshore during your stay, you may get backwash rain.
We go in late September most years, and usually there is only the occasional brief thunderstorm. The parks have only had to close due to hurricane one time, for Hurricane Floyd, Sept. 14-15, 1999.
There is another possible drawback to be concerned with in September, round 2 of the lovebug mating cycle. This usually carries over from late August into early September, so the latter half of the month is generally fine. Lovebugs don't bite, but when they are in cycle they are constantly flying in the daytime, and it's a nuisance.