Carpooling: Consider taking the bus or a taxi into Seattle from the airport and carpooling with other guests to the wedding! Some hotels also have shuttles. You really don't need a car in Seattle, as it is very compact and bus-able. Let me know if you might be interested in carpooling if you are unable to connect with someone.
Train: Amtrak runs from Seattle to Wenatchee, 25 miles from Leavenworth.
Bus service: A bus runs from the Amtrak Seattle station and the Greyhound Bus Station in downtown Seattle for Leavenworth twice daily. Northwestern Trail Lines runs the service for $26 one-way or $46 round-trip. See the link for the schedule.