Wyoming's climate can fluctuate from the high 80s to the 40s, so take a lightweight down jacket or vest. If you're sensitive to the sun, pack long-sleeved cotton shirts and sunscreen. By mid-morning I usually shucked my long-sleeved shirt and rode in the sleeveless t-shirt I wore underneath. Rainfall is only 12 inches a year, but take non-bulky rain gear just in case. Don't forget your bathing suit.
Outfitters usually provide tents and thick foam pads; guests bring their own sleeping bags and towels.
Helmets are not required on most Western rides, and most of us rode in cowboy hats or baseball caps for protection from the sun. Make sure your hat has a cord so it won't blow away.
Water on the trail is a necessity, but don't buy one of those western canteens. They look good, but it takes two hands to hold one and remove the cap, and the water tastes metallic. A large biking water bottle is ideal for one-handed drinking and is easy to tie to the saddle horn.
A fanny pack is handy for carrying small items such as sunscreen, lip balm, eyedrops, a pocketknife, or a bit of string in case something breaks.
By all means, pack light. You'll be lugging your bag (make it a soft-sider or duffle) to the chuckwagon every morning. Because the humidity is low, you'll need fewer changes of clothes than you think. A sweatshirt and sweat pants will do for nightly camp wear.