I took my nephew when he was 16 to Europe for two weeks. It was a wonderful experience for both of us. I had to establish some rules prior to our trip and rules during our trip regarding spending and itinerary changes and other such thing, but it was great. I found that he was very different with me than he was with his father, much more well behaved and listened and really found himself. I did not mother him but I did not sit back and let him do whatever he wanted (well, other than on our last night allowing him to just get room service rather than experiencing one last night out in Europe). We did what I had laid out for an itinerary and we accomodated any special interest either one of us had. He tolerated going to the caves with me and I tolerated biking umpteen million miles through the Loire for him.
This probably isn't relevant to taking children to Disney World, but I think the bottom line is that it was one of the best experiences for me and for him and we got along so well during that time and he was absolutely not a problem at any time during the trip. And just so you understand where I am coming from my nephew is not a peach, he and I clashed all the time when not on vacation together as I was a sort of mother figure for him even though I lived several thousand miles away. He also has extreme issues with authority. So that being said, he and I had a wonderful experience together.
Lay out ground rules for stuff that you know may be an issue like sweets, merchandise and bed times and all will go so much better than you ever could hope for. I hope you have a wonderful time doing this for them.