Fly into Bozeman Yellowstone International Airport (BZN):
Bozeman Yellowstone International Airport BZN | Yellowstone Flights (bozemanairport.com).
The Bozeman airport has recently expanded (again!) and serves major airlines, including American, Delta, Southwest and United, as well as several smaller and discount-fare airlines.
Follow the rainbow!
The common venue for both the wedding and reception is just off US Hwy 89, which runs north-south between the North Entrance of Yellowstone National Park to the south and Livingston, Montana, to the north.
GPS will do a good job of getting you to 63 Summit Way (Gardiner, MT), which is technically the address of a 2-story, yellow office building at the tranquil and picturesque Royal Teton Ranch.
The chapel for the wedding is right across a small green lawn from the office building. Called King Arthur's Court, the chapel can be identified by its steeple with stained glass and a shiny gold-toned top.
Turn-by-turn:
From Bozeman:
Go east on I-90, then take exit 333 at Livingston to get onto US Hwy 89 south (where it starts as Park Street) toward Gardiner and the Yellowstone National Park. You'll drive south on Hwy 89 for somewhere between 45 minutes to an hour. Take your time (slower-moving traffic on this rural two-lane road may force you to). Enjoy the wildlife along the road, especially south of Emigrant, and just stay alert to not hit any!
Stop at neighborly Emigrant General Store (https://www.yelp.com/biz/emigrant-general-store-pray) for gas, groceries, supplies and refreshments, and visit the tiny Meditation Point travelers' chapel where Gregory proposed to Dominique (successfully!):
https://www.yelp.com/biz/meditation-chapel-emigrant.
For an even more scenic journey, detour through East River Road, which runs north-south, roughly parallel US Hwy 89: https://scenicusa.net/032712.html.
Continue on US Hwy 89 through Yankee Jim Canyon and turn right on Cinnabar Basin Road at Yellowstone Hot Springs. Cross the Corwin Springs Bridge, which famously survived the flood!: https://www.youtube.com/watch?v=A5HoMcdgxW0 ; Floating house hits Corwin Springs Bridge in Yellowstone (youtube.com) (https://www.youtube.com/shorts/-VQ_EjzdXVI)
Take a right immediately after crossing Corwin Springs Bridge, follow the dirt road and when you arrive at a three-prong fork, take the center option in order to continue onto Summit Way. The road will bend left and you'll arrive at the 63 Summit Way office building on your left (you'll also see a yellow barn on your right). Turn left and drive past the office building for parking by the chapel or follow the horseshoe oval for parking along the nearby road. (See below for more on parking.)
From Gardiner/Yellowstone National Park:
Go north from Gardiner on US Hwy 89 where it starts as Park Street. Drive about 8 miles and turn left on Cinnabar Basin Road at Yellowstone Hot Springs. Cross the Corwin Springs Bridge and follow the remaining "From Bozeman" directions above.
Abundant free parking within easy walking distance is available all around the chapel, by the cafeteria or office building to either side of the chapel, or along the surrounding oval.