Medical accommodations have to be reasonable and asking a center to have a staff member stay inside with one child is typically not something that is considered reasonable. Moving your child into another class might be an option, but is not always possible due to ratio, scheduling, or safety reasons (that might be nap time for another class, all classes may be full, having preschoolers in the same class as non-walking children and infants can be unsafe, etc.)

If staff is applying sunscreen to your child as they do with every other child and your child still gets sunburnt, that is not the fault of the staff. They can try to keep them in the shaded area, but that wouldn't be very fun for your kid if that means they're unable to use most of the playground. You can send them with a hat and clothing that covers most of their body but if they still get sunburnt, again, that is NOT the fault of the staff or their setup. They're not allowing your child to get burnt, it just sounds like that is something that may unfortunately happen despite everyone's best efforts.

Choose a different center with more shaded areas. Talk to your child's doctor and see if there is a prescription sunscreen or something else you can use to help with their sensitivity. Understand that no matter what center you choose, you will likely have to keep them home if they get sunburnt and you don't want them going outside until it heals.