Traveling west on US Rt. 14 from Dayton, WY, you have three campgrounds in the Burgess Junction area. They are Sibley Lake, Prune Creek and North Tongue. Continuing west on US Rt. 14A, there are Bald Mtn. and Porcupine campgrounds. Don't forget, while continuing west on US Rt. 14, you travel through the Shoshone National Forest too - check out the campgrounds out of Cody and Wapiti.
For Bighorn campgrounds, click here http://www.forestcamping.com/dow/rockymtn/bigh.htm
and here http://www.forestcamping.com/dow/rockymtn/shos.htm
for Shoshone campgrounds. A thought for you to consider is there are 75 national forest campgrounds surrounding the Yellowstone and Grand Teton National Parks. We encourage folks to research these campgrounds as places to stay and visit the parks from there. I guess we are biased, but the experience is soooo much better.
Regarding your question about reservations on the 4th of July weekend, if your selected campground takes reservations, yes, yes make a reservation NOW
. If a campground doesn't take reservations, getting there no later than Sunday or Monday prior to the 4th might get you a campsite. Information about whether a campground accepts reservations (and how to make them) is available through the above links.
Forgot to mention, Cody has a great celebration from July 1 -4 every year - rodeo, parade and other kinds of traditional western events. We had a great time! See http://www.codystampederodeo.com/