Most churches, fire departments and town parks probably won't care if you have a van, and probably will be okay with the dogs, although they likely won't let you bring them inside.
Private homes and Warm Showers hosts will be on a case-by-case basis. Be considerate and make sure you call ahead, tell them about the van and dogs, and give them an easy chance to say no. Some of these people may feel that a van-supported group doesn't need their support as much as others, and many are not equipped to handle your dogs.
Hiker/biker sites in Yellowstone and Grand Teton are not available to those with a vehicle. An individual site will be cheaper for you anyway than seven times the per-person rate, although you face the possibility of a full campground. Send the van ahead early to reserve a spot.
I would guess that 90% of the cyclist-only camping spots will still be available to your van-supported group. You might miss out on a few more because of the dogs. You can send the van ahead to check things out. With seven of you, motels will become a pretty inexpensive alternative.