Is it legal to charge the money for application at the summer camp?

Yes, this is perfectly legal. It is a processing fee. They probably do a background check on everyone who might come in contact with the kids. They have to do that to protect themselves from potential law suits from parents who take their kids there. They must use reasonable precautions to keep all the kiddies safe. It costs time and money to do checks and frankly, $50 is pretty reasonable.

Landlords are legal to charge fees to process tenant applications too, for all the same reasons.

Edit:

No there is no legal reference. It is legal until there is a law that makes it illegal. That is like saying give me the legal reference why it is legal to wear a blue bow in your hair.

Even if you sign some form saying that you will not hold the camp responsible for any injuries, you can still sue if they were negligent at something. They still have to use reasonable and prudent precautions to keep the kids safe and they can be sued if they don't.

Bottom line, you are not being forced to choose that camp. You are not being forced to pay the money. It is your choice to "purchase" the product that they offer. You have the legal right to say no, I choose not to pay that fee and I choose not to allow my child to participate in your summer camp.

Yes, they can charge whatever they want for the application fee. You know that on an application you do not always get accepted.