There is actually a couple pages devoted to the mosquito thing in the UNOFFICAL GUIDE if you happen to have one to look at. I grew up in FL and can verify that everyday is mosquito season. I mean there are better and worse days. (The warmer and more humid the worse they are, the cooler, the lesser the activity in my experience). But according to the above mentioned book Disney goes to great lengths to spray and keep the little bugs under control. I have only been bitten a couple times when we have been there and it seemed to be in the August trips. Mosquitos are bad everywhere anymore...I live in the NE and we have to now content w/ West Nile and ours dogs have to be on preventive heart worm medicine all year long (even now at 30 degrees!) but I'll have to say Disney does a wonderful job of keeping it under control.
Chiggers you get usually from sitting on the ground or in high grasses/weeds. If you avoid sitting in the grass you shouldn't have a problem. I don't know if bug spray works...as a kid we never used it.