HIPAA guidelines state that health information is on a need-to-know basis. In this case, your adventure staff need to see the health information in order to do an effective screening. It is not a problem. (I'm curious as to how anyone might think it IS a problem?) Of course, any camper or their parent can request that the health staff do their particular screening. I doubt that any would, unless they have health concerns--but it would be wise to have anyone with real health needs be screened by the person with the highest certification, anyway.
Definitely, you would need to educate your screening staff--particularly non-medical-professionals--on privacy; make sure they understand that information gathered during screening is not to be passed along indiscriminately or misused.
That said, I don't honestly find it that hard to screen ~120 campers with two people--one doing the primary screening, one doing a lice check. Unless you're a special needs camp, I wonder if your process needs streamlining?