as someone who has battled shin splints and had 2 bouts with shin stress fractures, I definitely get where you're coming from.
What can help greatly is having additional arch support, either through custom made orthodics from a podiatrist or even off the shelf orthodics - that can make for a huge improvement! - put them in whatever shoes you are wearing that day. Granted, that eliminates the ability to wear flip flops, but it's better than sore shins! I haven't put them in crocs, but I don't see why it would be a problem - if you put them in and find that they slip out of place, a couple pieces of double stick tape on the bottom of the orthodic should keep it nicely in place for the day.
Also, if you are in a resort with a fridge (that I presume has a freezer) bring paper cups with you, fill them with water and freeze them and use them to ice your shins for 15-20 minutes at the end of the day. I have also gotten decent relief by taking some ibuprofen as well.
Like the previous responder said, it is caused by the way your foot strikes the ground, however it is not always something that can be fixed just by altering your foot strike pattern - and quite frankly, after you've been walking around for several hours at Disney, you're going to be too tired to focus on your foot strike pattern! Look into the orthodics and if need be, look into icing. Overall, if you know you have shin problems start training for Disney - build up your walking tolerance in the time before you go and doing exercises to build up the strength of the muscles in your lower legs. The stronger the support system there, the better they will feel. There are a lot of simple exercises that you can do with an exercise band that will help.