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The idea of getting local business involved is a good one. Most are interested in helping kids.
What has worked best for our church is to ask the congregation on a local church to support their youth going to camp.
Present a slide show during the church service that tells just how great this will be for them. Then ask the congregation to contribute outside of their regular pledge. Have envelope ready marked "Send a Kid to Camp for two weeks. It will benefit them for a lifetime."
Have been very involved in Christian Youth Camping virtually my entire life. I would echo Tony's thoughts about tapping into the church's that have been active at the camp. I assume the facility is also used for rentals - so some of those groups that have had good experiences at the camp could be good resources for sponsorship donations.
I would also take a look at where the current paying campers come from. In my experience, there is often clusters of common church's that these kids come from. They would also be good prospects for sponsorship underwriting.
Lastly, I would look at some web resources like GoFundMe. This a crowdsourcing site dedicated to raising funds for philanthropic and benevolent causes. I would definitely post there. But you have to create a compelling story - but hey, you are in advertising so that should be the easy part!