Summer season officially begins the first or second week in June and peaks during the 4th of july and most of that month. The worst of it would be Xmas week but summer crowds are much more manageable, but still... expect crowds to some degree.
also if you haven't been to florida during the summer, expect it to be hot and humid. as was said above the park is crowded during the beginning of june but not wall to wall people as many northeastern schools don't get out till the end of june. the "official" summer season doesn't start untill the end of june and runs till beginning of august , but things begin to pick up crowd wise after memorial day.
Summer crowds begin to roll around about mid June, but July is usually the busiest month of the summer and usually the second busiest time of year (the first being Christmas). That's not to say that everything will have 70 minutes and zero manueverability on the walkways, its just going to be a little more crowded than usual. But going then in June, you should be ok, but expect there to be rain practically every day at least for a few minutes.