AS crazy as the weather has been here in Texas this year, I can only guess about Florida...
but "normal" June weather is very warm and humid (HOT to people from the North) and clear to partly cloudy, breezy, with afternoon thundershowers, occasionally severe.
Dress for warm weather but have a rain poncho handy.
When we were there in June of 2000 it was blazing hot and hardly rained. Last June it rained almost everyday, all day long. In some ways, the rain was welcome as it wasn't so HOT! I could have done without the thunder storm though at AK that knocked out the power.
I know in 2000 I sat in the sun holding a spot for the afternoon parade at MK and then we flew over to MGM and I was over heated. Be careful and drink a lot if it's hot. I made sure the kids were okay and forgot about myself.
It's not bad though and it justifies having floats
Plan on Hot and humid. We were there that same week in 2002 and had many hours of rain, every day for the whole 10 days.
Last yr went a week earlier and only had rain a couple days and they weren't the total drenchers we faced the yr before.
But we didn't learn our lesson...we'll be there the 13th - 21st this yr.