All the ice-cream stands in the parks sell both a strawberry Minnie bar (juice bar) that is dairy free and a Popsicle called an isacadozi that the kids love. My older DD will choose that over a Mickey bar sometimes and she has no problems at all with dairy.
DD was fine with the pop corn at all parks too!
It is very hard to find a dairy and egg free treat in world show case. The shaved ice in Japan and the plan pretzel in Germany (nasty dry if you ask me) were all we could find. We bring our own treats for DD for times like that.