My kids are these same ages, and I often find myself in a moment like this — wanting my boys to feel happy and successful, and wanting to help make that happen. The trouble is that at 8 and 10, you don’t have much control. Your kids will probably handle camp the way they handle other experiences. If one of them is often anxious or shy in new environments, that will probably happen this time too. If the other is good at making new friends, that’s also likely to occur. In the end, this experience will probably have more to do with slap-bracelets and swimming tests than Judaism.
That said, in my opinion, the best tool you can give them is the truth — a clear sense of how your family is intermarried and proud/happy that way. I’d explain that they’re heading to camp to learn, and that THAT is inherently and deeply Jewish, the desire to learn. I’d task them with learning, since that’s the aspect of a new environment they have the most control over. I’d ask them to bring home something new and interesting for you to learn each day too, so that you can share this experience with them.
The other thing I’d do is to reach out to the camp director, and voice these feelings. If the camp is inclusive (as you say) this is nothing new. Likely, you’ll be quickly reassured that a lot of their kids come from homes like yours. Maybe you could even be connected with another family for carpooling or something!