Will my journey to intern at a summer camp be allowed under the US Visa Waiver Program?

From the US embassy in London but presumably applicable to your situation:

If you wish to pursue practical training through an internship with a U.S. based employer you will require either an exchange visitor (J-1) or trainee (H-3) visa. Such activities cannot be conducted on a B-2 visa or visa free under the Waiver Program, even if you will receive no payment from a U.S. source.

For it to be "volunteer" work, which is in some cases covered under the VWP, it must be work that is normally carried out by volunteers, for example see this question. That is, if anybody doing the same work as you is paid, you cannot do that work under the VWP, regardless of whether or not you are personally paid for it.

Here is another US embassy reference that emphasises the fact that no work of any sort should be carried out on the VWP, even unpaid work, work in return for room and board, etc:

Unpaid Work is Work – Make Sure You Have the Correct Visa (March 10, 2015)

Planning to work for room and board on a farm or as an Au Pair/Nanny? Do Woofing? An internship? Even if you stay less than 90 days, these activities are considered work and require the appropriate visa. Visa Waiver Program travelers entering the United States with ESTA authorization are not allowed to work. If you are a Summer Work Travel participant (J-1 visa) you may only work in jobs approved by your designated U.S. sponsor. Informal arrangements to work in exchange for lodging or meals are also considered unauthorized employment and are not permitted for tourists.

I don't know that there's anything you can do in such a short time scale. If this company did not provide you with appropriate information, documentation and advice to allow you to get the proper visa, you probably don't want to be working for them.

Doesn't sound legal under VWP. J-1 is usually used for that sort of thing.