Disney has a contract with Actors Equity through 2022 which requires that certain roles (basically exactly as
@DoleWhipDrea said, mainly scripted speaking/singing roles or choreographed roles in stage productions - but not parade or M&G performers) be filled by Equity actors. It is possible that actors who are in AEA can fill roles that are not part of the contract, but I'm pretty sure the Disney contract requires that those roles then be paid at equity rates - which means that it almost never happens! If Disney was to alter/rename roles and recast them with non-Equity performers they would be looking at a labor dispute that could have HUGE ramifications for Disney (SAG and AEA are sister organizations, so movie production could also be greatly impacted if AEA feels that Disney is breaking their contract). For many actors (in fact, every actor that I know, but I'm hesitant to put out a generality like 'most'), getting to the level of being in Actors Equity is a proud achievement - you have to actually perform at certain level of quality theatre for a specific length of time to be allowed to apply for membership. If WDW was to cut off their ties with AEA, the pool of high quality actors would certainly be reduced. And, I for one, am consistently impressed with the quality of actors in the shows at Disney and would be very sad to see that change.
Universal stage performers have voted a few times to not become a part of Actors Equity, so they don't have the protections of AEA - not sure about testing at Universal for performers. Not positive when the last vote was, but I am almost positive that AEA does not have a contract with Universal in Florida.