Upvote:2
What I think you are looking for is commonly known as a rideshare or carpool, i.e. a ride in someone else's private car where you might compensate the driver for expenses. Unfortunately, because Riverton is a small town in a remote and unpopulated area, I think it is extremely unlikely you will be able to pair with someone driving all the way to Seattle. There are likely to be people in the area who make trips to closer cities however, among them Jackson, Caspar, Billings, or Salt Lake City, which have better connections.
Some services that try to match drivers and riders include Rdvouz.com, Ridesharing.com, and Kangaride; I have not used any of them and do not vouch for them, but they are worth a try. The best-known name in this sector was Tripda, but unfortunately they shut down in early 2016; Ridejoy is also long gone. Zimride remains, but you must be affiliated with a participating institutional partner (e.g. employer, university) to use it.