Depends what you mean by cheap.

The cheapest option will be camping. Moran State Park is the only public campground. If you’re going in the summer you need reservations, if you’re going in the winter you might have the whole place to yourself.

If you’re going now, which is the off season, hotels and B&Bs are much less expensive than in the summer. I don’t know where the lowest prices would be. The only place I’ve stayed on Orcas is the Turtleback Farm Inn, which is very nice (sheep! fireplace! port and cookies at sunset!) but rates there start around $100 so I wouldn’t consider that cheap.

Depending on the season, Outlook Inn offers rates starting at $54 for European-style rooms with a shared bathroom.

You can camp at the base of Mt Constitution, rent a Yurt for cheap, or rent a trailer hotel thingy in Doe Bay for a little more.