Nothing at all to worry about.
Only 40% of the WDW workforce is represented by the unions. Of those, many may choose to cross picket lines if a strike were called--43% of the union workforce voted IN FAVOR of accepting the new contract.
You've also got a significant portion of the 50,000+ workers at WDW who are currently part time. Their hours could easily be increased if necessary.
Also, with Florida being a Right to Work state, there is no mandate that eligible employees join the union. This makes it fairly easy for Disney to hire temporary (or permanent) replacements for striking workers.
And, as long as a Federal Mediator is working productively with both parties, no strike will be called.
Right now there's a lot of chest-thumping going on by the unions involved. There are multiple unions simultaneously involved in the negotiations with Disney management. When the tentative agreement was reached last week, reports indicated that a couple of the unions were not represented at talks. Even if the deal was fair, the unions feeling snubbed certainly got their membership riled up and recommened they vote against the contract.
The first two proposals were declined by the following margins:
4155-15 against
3659-96 against
This last vote was 3686 - 2827 against.
Sounds to me like they are making progress. Hurt feelings and bruised egos are going to prolong the process. Eventually they will reach an agreement.
In the meantime, life will continue at Walt Disney World.