I run residentials, but it is with a group that knows the leaders well. The trips tend to be about 3 days and the longest we did was 5.
Homesickness is common and tends to come on at night when children are tired. Heck, I've had it as an adult on the other side of the world after a couple of weeks, when patience had been exhausted with a somewhat "trying" travelling companion and period pains came and finished me off, and the time difference didn't work for phonecalls to DH. 24 hours later I was fine again and ready for the next 10 days.
In this case try and get hold of the leaders and get them to talk to her.
It may well be that having expected to go with friends, it's a disappointment that they've dropped out and it is hard to slot socially with other clusters.
If it is deeper than settling in issues (and getting into the routine of the set-up helps a lot) then have a plan to pick up in a couple of days. It's not an immediate exit, but it eases that pressure of a full week and she'll have a better chance of finding some positives.
You are still being trusted and listening to her, not dismissing her, but patience is a compromise.
Ploughing in for an immediate rescue is a safeguarding issue. There may be issues with unplanned access to the site, and it will affect leader supervision issues while the matter is dealt with. She may well not be instantly avaliable anyway, so liason with staff is essential whatever the outcome.
She's not much younger than DoEers who find themselves in the wrong field squabbling about it while limping around with blisters. I found that a valuable experience in learning that sometimes you do have to get your head in gear and plod on out of the situation; and often in hindsight it wasn't as bad as it felt in the thick of it.