Not a strange question at all! (I also had one that went through a phase where he would only nurse in the side lying position, although we luckily didn't have a Disney trip planned during that time!)
What I did (and would suggest) is to take along a blanket (bigger than a "baby blanket", but not something huge) that you could put down on the floor. That worked fine for me. The nursing rooms are nice, but I really only used them once or twice. They were too far out of the way for me to make use of them more often than that. Too inconvenient when there were plenty of nice benches, shows, rides etc. that were great for nursing. But, in your situation, it sounds like the nursing rooms will be a great help!
Definately don't resort to pumping. That's a HUGE amount of work if you don't have to do it.
Have fun!