You might just try "dog sitting/ local zip code" on Google, and see what you get.
I doubt that you'll find someone at a particular campsite, though you may find someone who is willing to watch your pets for you, and even walk them once or twice, but you are asking a lot of people you have just met....and your dog may or may not be comfortable with this either.
Personally, if we were going to be gone for 6 hours or less, I think I'd have the dogs crated up in the RV, with the A/C on. I would probably suggest to a friendly neighbor, that the dogs are inside, just so they know, if the event of a problem, but I don't think I'd expect them to do anything, nor would I expect them to go into the RV. More than 6 hours, without returning, and I would probably seek out some sort of kennel or Doggie Day Care for the day.
Bill & Claudia / DD Jenn / DS Chris / GS MJ Dogs: Sophie, Abby, Brandy, Kahlie, Annie, Maggie, Tugger & Beau RIP: Cookie, Foxy & Gidget @ Rainbow Bridge.
2000 Winnebago "Minnie" 31C, Ford V-10
Purchased April 2008 FMCA# F407293 The Pets
Ask at the office. They will know if anybody is available. We have always found somebody when needed.
2007 Northwoods Arctic Fox 32 5S Fifth Wheel used for fulltiming for several years--SOLD
2011 Keystone 23rks Hideout to poke around the smaller parks in the great Southwest
2007 Chevy Silverado 2500 HD Diesel
Prodigy brake control
A 'hostess' at Riley Creek did this service one year for nominal amount. A 'ranger' at Mendenhall Glacier (in Juneau) volunteered when we asked her how we could find a dog sitter. Had another person volunteer when we signed up for an all day boat cruise out of Seward. Normally we leave the dog for 6 hours without a porblem; we do kennel our dog for overnights and usually when we go into Denali for 3+ days.
I always check out local kennels BEFORE we leave our pet and some did not meet my requirements so we made alternate arrangements.
Usually there are kennels fairly close to the places I've camped but generally I don't bring the dog if we are planning day trips. Because my dog gets along well with other animals, when I have been approached by a camping neighbor, I have offered to see if their pet gets along with mine to have them hang with us at the campsite. I've also offered to go check on them or take them for a potty walk in a few hours. Been taken up on it once or twice, most said their animal was fine in the trailer and just want ed to let me know in case of an emergency.
2011 Silverado Crewcab 4x4
2012 Passport 238ML
Hope your travels are safe and the friendships made camping are lasting.