Wyo West Lodging is a local rental company. Be sure to filter on homes "in town" to be close to the wedding, shops and entertainment,
United is typically the best option for flying into Cody. The connection are thru Denver. If you are true Southwest flyer than the next closest airport is Bozeman Montana (3 hours away). Cody airport is a 10 minute ride to downtown and the wedding venue. The airport offers car rentals but we have used Turo rentals and enjoyed the experience.
Turo car sharing is a fun way to rent a variety of vehicles directly from the owner. You get remote access via the app when you arrive and there are affordable insurance options. Rent an older vehicle and save some money or rent that luxury ride you have been thinking about. You can pick up at the airport or save the airport fee and pick up at your hotel.