I've been most of my trips in July, August, and September.
Whenever you have high temps and high humidity, there's always a chance for that kind of weather. It happens usually between 3-7pm, and as soon as it comes in, it's gone. Or it seems like it moves quickly. I have never been in those months where it rains non-stop all day. It lasts maybe an hour most times, but I've seen them last longer occasionally.
I poncho up and keep going. I'm not letting rain stop my day; if it is absolutely torrential, I will pause and spend an extra five or twenty minutes in a gift shop.
My mom said she and my Dad got on Small World and just rode around and around to ride out a September storm. (It was the mid-80s and evidently not as busy as today.)
Get a poncho or a raincoat. Umbrellas are not great with lightning; then again, neither is being outside. You'll be fine.