Hi Lisa,
Absolutely! In most cases, the parks are open rain or shine. My best advice to you is, don't let the rain slow you down! While most guests probably do not want it to rain on their vacation, my experience is the rain thins out the crowds at the theme parks, especially if it is more than just a passing storm. The skies open up and people leave in a mass exodus. I, on the otherhand, pull out my poncho and enjoy the short lines! Most of Walt Disney World can be enjoyed in the rain and ponchos are available throughout the theme parks.
My family just keeps on going. One of my greatest Disney memories was in the Magic Kingdom one evening. It was torrential rain and people were hiding undercover everywhere they could find. My crazy kids refused rain ponchos and instead they went into the middle of the street and started dancing in the rain. A Cast Member brought hoola-hoops out to them and they had a blast. We often wear flip flops and sandals too, in case of rain. They do sell rain covers for strollers, so those don't get soaked.
There can be some rides that do close, but many just keep going. Epcot is a great place to be when it rains because many attractions are indoors. Nearly all of the character meet and greet areas are enclosed, so rain shouldn't keep you from meeting them. Unfortunately, the parades are sometimes cancelled to keep the cast members safe, but I have seen those in the rain as well. It depends on how bad the weather is.
Overall, don't worry about the rain. Pull out your poncho have a blast!
Kaylene