It will take longer with a wheelchair to seek out shelter. We did it with a wheelchair for about 5 years. A poncho is probably the best idea. We watch the forecasts every day very carefully to get a good idea if morning, afternoon, or evening rain showers are likely. Based on the forecasts we plan our day accordingly. For example, if showers will be more likely in the afternoon then we concentrate on the outside stuff more in the parks in the morning. When it starts raining we concentrate on inside activities. In Epcot, for example, The Land Pavillon is great to be in during a storm. Just remember, that everyone else will be doing the same thing, here; so, The Land will be crowded.
If the park goes from sunny to overcast we get ready to seek shelter. See a show like the American Idol at Hollywood Studios. We look at the sky a lot and listen for thunder if rain is in the forecast. Shelter can be an inside show, an inside ride or an eatery. When waiting for a parade or fireworks, we will be near an inside attraction; so, we can seek shelter quickly that way. At MK, the Hall of Presidents area is a great one if it starts raining during a parade. Also, the worse place to be for a parade would be in the hub by MK's castle. There is no way you can move from this area quickly enough without getting very wet if it rains. You need to be in the alert mode and you will be fine. At the first sign of a drop of water from the sky we seek shelter.
We got caught once in a bad storm (no forecast for rain) and did get drenched. All you can do is go back to your resort room and change clothes and dry up. We didn't even have our ponchos that day. Maybe, just carry your gear all days you are there. I don't recommend drying off the hard way; don't stay in wet clothes. You can take a change of clothes to the park if you are really worried about it.
If it's raining before we start our day we stay in our room. We will wait for the rain to end.
