It's not a blanket yes or no. While the staff for the school and camp are often seperate, in some schools you will see some coaches and teachers working at the camp during the summer months. Now, my DS is much older but "back in the day" the scoop for Landon was that is was very beneficial for you son to attend summer at Landon because the AD was heavily involved with summer camp and it was a good way for him to get to know your son. As such, getting your kid into summer at Landon was a feat unto itself.
So, I would look closely at the camp associated with the school you are interested in. If nothing else, it will let you get familiar with the setting and some of the people and while it may help, it will in no way hurt. You can then tell the AD "we just LOVE Whatever School so much that Jr. has been coming to summer camp here for 3 years, such a lovey experience, blah, blah" which played correctly will not hurt.
Please, no flaming about Landon, I know how many of you look down your nose at Landon, I was using it as an example of cross pollination between summer camp and school which is what OP was looking for.