You can send food that needs to be kept cold; you just need insulated lunch bags and freezer ice packs (both are sold in most grocery stores now). I use them all summer long for my daughter's camps, our own car trips etc.
If the camp insists they must bring food in a brown paper sack and/or says all lunch materials must be thrown away (in other words, no lunch bags, lunch boxes or containers that need to come home each day), I'm sorry! Try freezing grapes and putting those in their lunches (they will be thawed but nicely cold at lunchtime), sending dried fruit, healthy snack bars (many have better nutrition than a "real" meal if you read the labels), etc. Be sure to freeze drinks in juice boxes etc. and they too will be defrosted but cold at lunch.
How hot does it get there? I would hate to send a peanut butter sandwich to outdoor camp with my kid here when it's 100 degrees all day in July here, and very humid....it would be gross by lunchtime even though PB technically "keeps" OK. It would just be warm and runny, around here.