Well, we went in May, stayed at the Fort Wilderness Cabins, brought along bug spray and it sat on top of the refrigerator the entire time we were there. If you are concerned, take something small but you probably won't have to dose yourself with it every day. My husband says that Disney wouldn't let a mosquito onsite. And he is a mosquito magnet.
And I *loved* the Florida Double Bug comment. That was classic!!!!!!