Are there many rentals available in Washington? What websites would be good for finding them?

Some, not many. On zillow.com and craigslist, a lot of the rentals (and I only count two dozen) are homes, and some of them being quite nice, they can carry extremely high monthly rental fees. There are a few apartments here but the town is so small to begin with that I can't imagine you'll find many more than I just did when I searched.