It could be a number of things. The touching? Check. No one wants to lose their job or license because of a child incorrectly explaining something that might have happened while applying sunblock. Some states might consider sunblock a medicine; at my preschool families are asked to sign a medical consent before I will even open up a tube of sunblock. I only will apply it to faces and arms, and expect my families to be primarily responsible in making sure their children are dressed appropriately and prepped for the weather every day.
For a preschool teacher in a larger group, it is a task of enormous proportions to slather up ALL the kids and make sure not to miss anyone. My program is small (4 kids) but it still takes a long time to do this. Kids aren't always great at getting themselves covered and not getting the stuff in their eyes. While I do ask parents to sunblock kids before school on sunny days, we live in Portland and the weather changes, so I like to have the option to be flexible. (I only will apply it to faces and arms/exposed parts.) I don't want any children to get burned on my watch.
As a parent, I am also very careful about what kind of sunblock my son uses. (We use an expensive, nonchemical one.)
For what it's worth, consider SPF clothes. They are wonderful. Carters and OP sell them for very reasonable prices. Sun hats, too, are a must. If it's too hot, kids should be provided some sort of shade or relief from the heat, and good providers will know when it's time to go in because there's no reason for kids to get bad sunburns in care. Mild sunburns happen, even with the best of intentions, but more serious burns shouldn't be happening with adequately trained staff. Period.
For what it's worth, sunblock does usually need to be reapplied every 2-3 hours or so, so I'd ask the director/teachers at your son's school and camp what their plans are for very sunny days. If it's a full-day program, there needs to be some back-up space for the kids to go to out of the sun. And don't be afraid to ask--you are the client and paying for the service. If they can't give you a satisfactory answer or help form a plan, it might be worth it to rethink. I'd never be upset at a parent for asking thoughtful questions about their child's care, and neither will any high-quality program. Best of luck!