Just to clarify what some others have said, you *can* do a rail-only trip, but as of today, it will require a one night stop-over in Seattle, as the Vancouver-Seattle Cascade arrives long after the Coast Starlight has departed. (There is talk of adding an additional train frequency from Vancouver to Seattle and vice versa, but no firm start date yet, and no guarantee that it'll leave early enough to connect with the Starlight).
Alternatively, Amtrak does provide a same-day, guaranteed motorcoach connection to the Starlight from Vancouver, but it leaves Vancouver at 5:30 AM.
Also keep in mind that if you take the motorcoach connection, the fare for the Vancouver-Seattle segment *should* be significantly cheaper (around $20 cheaper, based on what I'm seeing when booking the itinerary via amtrak.com) than booking two separate segments for a rail-only (overnighted) connection.
-Rafi