As long as the puppy isn't licking/chewing on the other dogs, it should be OK to treat them. I'd recommend keeping puppy away from them for a day so the chemicals can absorb as much as possible.
I've been hesitant to use any flea treatment within a week of inoculations, but as you're going camping in a tick area, I would check with your vet to see what they suggest. I don't use Frontline - found it ineffectual for fleas - so I can't answer to that part.
I do use Advantage/Advantix on an as-needed basis. For fleas, the great thing about Advantage and Advantix is that you don't have to treat every pet in the household. The fleas jump on and off the dogs, so the fleas on the untreated dog will visit a treated dog and get zapped. For ticks, it's not so convenient. I'd recommend a good tick tweezer and a thorough check of the puppy at least twice a day (at bedtime and around noon). I'd also check him after any hiking or rolling around.
Another option you have is to use Zodiac household spray on the puppy's bedding. Spray the bedding a day before you leave so it has a chance to completely dry. I usually spray one side, then flip it so that's on the bottom and not next to the animal.
1975 American Clipper RV with Dodge 360 (photo in profile)
1998 American Clipper Fold n Roll Folding Trailer
Both born in Morgan Hill, CA to Irv Perch (Daddy of the Aristocrat trailers)