The Washington State Ferry is the most practical and least expensive way for most cyclists to reach Orcas. The ferry system runs car/passenger/bicycle boats to the islands from a mainland terminal in Anacortes, about 80 miles from Seattle.
To get to the Anacortes terminal from Seattle, take Interstate 5 north past Mount Vernon and exit onto Highway 20 at Burlington. Highway 20 (called Spur 20 on the signs) takes you straight into Anacortes, where you can follow the signs to the ferry.
Reservations are strongly recommended for travelers taking a vehicle to the San Juans. You can make a reservation both ways. Go to Take a Ferry for more info and to make a reservation.