WDW (as do many other fast food locations) provides little paper cups that you can put cheese/catsup/mustard/mayo or anything else you want into.
Remember those little pleated paper rinse cups dentists used to have? They are like little versions of that. They are all over the place, including near the cheese.
Many many many people fill them with hot cheese and put them on their tray. I have done it myself. Here is one with ketchup in it.
Still, I am of the opinion that when you fill the cup with cheese you know it is hot (you can feel it). If you then have a four year old, you do not put a tray (with cheese or otherwise) in such a position as to permit them to spill it on themselves. If you do, then it is your fault.
-dave