Personally, I'm always reluctant to allow our kids to wear sandals when we are touring around, as much as they love them. I used to sell shoes in a sportswear store when I was younger and a good pair of running shoes will be much more comfortable in the long run, provide much more support and, IMHO, give them more energy at the end of the day.
Also, I am always worried about much bigger people stepping on their feet in crowds or having people run over their feet with strollers or, like what happened to DS during vacation last year, getting his toes caught caught in the space between the floor and the bottom of a heavy fire door in our resort as he was opening it...ouch! I'm not sure how much better having shoes and socks would be in any of those situations but it's got to be better than sandaled feet. I'm probably a little more paranoid than others on this topic because both my DD and DS have "lost" (and regrown) the nails on one of their big toes when they were much younger through various mishaps.