The only sickness I've seen at WDW has been my own or my family's , thank goodness since I do NOT handle it at all. Seriously, if DH is sick, I leave the house and he will come outside and tell me when he's done because if I have to hear it I will be right there with him. Granted I have GERD and a hiatal hernia so vomiting is a fairly common occurence for me personally but I can't stand to see/hear/smell someone else doing it.
I got sick on my honeymoon on the way back from the Luau at the Polynesian. I know this should go without saying but DO NOT DRINK THE CHEAP WINE THAT IS INCLUDED WITH THE LUAU!!!!!!!!

I drank two glasses with my dinner, and I didn't really eat that much so the combination of not a lot of food and cheap wine and the FL heat in September had me VERY sick. I started not feeling good waiting for the bus from the Polynesian to DTD. Got almost all the way to DTD on the bus and I dumped my purse in DH's lap and threw up in it. The purse, not his lap. The funny thing is, aside from DH, no one noticed. I have no muscles in or around my esophagus so I am a quiet vomiter. Good thing I guess. We got off the bus at DTD and DH threw away the purse. I laid down on the bus stop bench for AKL. Very classy in a dress but I really felt bad. Threw up at DTD a few more times and then there was nothing left for the ride back to AKL, THANK GOODNESS. Heck of a honeymoon story LOL. Lesson learned and we will NEVER drink the cheap wine again!!
Our last trip this past August there was quite a bit of sickness but fortunately it was mostly in our hotel room. Both DD's got sick during the night and housekeeping was really awesome about quickly bringing new sheets and towels and blankets. They offered to call someone if we wanted but DD's luckily were not that sick. Little DD2 got sick at Donald's Breakfastosaurus. That was special. I am so glad that we were pretty much the only ones in our section and in addition to the
DDP tip of 18% I left an extra $20 since no one deserves to have to deal with that but they shooed me away from helping to clean up. The manager also came and brought me some food to go since I didn't get to eat at all and I had meds I needed to take with food. How great is Disney?

Oh, and Guest Services at AK gave us a voucher to come back another day since we were only in the park about 30 minutes before we left. DIdn't get to use it this trip, hopefully it is still good next time around. DD's luckily got the mild end of that stomach bug and I was the one who got it next and really bad too. No public exploits other than almost fainting and then crying at Epcot before DH made the executive decision that I needed to go back to the room. I have to say though, that the CM's are so wonderful to deal with. I ended up with Reedy Creek Ambulance at my room that night and DH says they were fantastic ( I was too out of it to know) and some guy in a suit from Pop Century. So I guess if you have to get sick on vacation, as much as it really stinks to be missing out on everything at WDW, they treat you so well.