Don’t let this worry you. It's that time of year when it rains every day. You'll find that, as most have said, it will usually rain very, very hard but for a short amount of time. There are times when it can rain longer but you still will have a good time. Again, do the outdoor rides when you know it's sunny and be flexible.
IMO, as
@Weather_Lady mentioned, wearing the right type of shoes is key. Wet, soggy shoes are extremely uncomfortable to wear for a long period of time so bring some that you know are made to get wet and dry out fast. Usually if you're in the rain, whether you have an umbrella or poncho, you will get wet. When it rains, it often rains sideways.
We are locals and we find forecasts are very unpredictable here. We have yet to understand what is meant by the percentage. Just this past week, days that had a higher percentage of rain didn't rain at all while days with a less chance of rain did. Plus, it might be that one area gets a ton of rain and other areas get none. For example, my husband rides his bicycle to work and there is a stretch of road that he takes that is a mile long and you can see clearly from one end to the other. One day, at the beginning of this stretch of road it was sunny and partly cloudy blue sky and dry as a bone but it looked hazy at the other end. He thought it was his eyes or the heat coming off of the pavement but when he reached the other end, he said it was pouring buckets.
When we're at the Parks and it rains, we take the opportunity to talk to Cast Members and learn a little about them since many are from other parts of the world. We also like learning fun facts and trivia from Cast Members and some will know and share design features that the imagineers have incorporated into the design of the parks that have meanings behind them. You could also purchase the "Hidden Mickeys" guide book and search for the Hidden Mickey's where ever you end up taking shelter.
As others have said, the silver lining is that the Parks clear out and you can squeeze in a lot more attractions in a shorter amount of time.