No, they are NOT peanut free zones. My younger child is allergic to all nuts and they do Ratatouille Cooking School in the Lab. The cookies they make are not guaranteed to be safe from nuts.
Additionally, I have read reports on here of parents of children with special needs having meals sent to the club for their children--including PBJ sandwiches. (I just read one such report within the past month.) PBJ sandwiches are not supposed to be allowed, but every time I've read about this, the meal has been allowed and the parents have been requested not to order that particular meal again.
I did not allow my child to eat in the club and it worked out fine for us, but your experiences might be different.