If anything, it would be pretty minor I would imagine. Can't see them closing down any attraction, especially one as iconic as SM in MK during the 50th anniversary year.
Also remember this with regards to the Disney IT department. The IT department worked hard on developing the system that is in place now. Just as it was being deployed, these same people were told that they were being laid off and replaced with contractors from India here on H-1B visas and...
Early morning sneak peak of the MK Holiday fireworks being tested this week.
https://www.wesh.com/article/disney-warns-neighbors-of-early-morning-test-of-new-fireworks-show/29549499
Ladies and Gentlemen. Boys and Girls. Due to a flock of birds in the park today, the Magic Kingdom will be closed today. These are just no ordinary birds, they are the most foul, cruel, and bad-tempered birds you ever set eyes on. Thank you and have a magical day.
I believe it's usually the Dept. of Health that tries to contact people due to HIPPA regulations. Disney will work with them to provide information as to lists of people who could have been exposed. Problem is that by now, people would have already left Disney and been in airports and planes...