I am a U.S. permanent resident. Can I get a Schengen visa in Seattle?

Firstly, you can't apply to just any Schengen country for a visa. You must apply to the country that is the main destination, or failing that, the country that is your first point of entry.

Secondly, none of the major European countries maintain a consulate in or near Seattle (that's France, Germany, The Netherlands, Belgium, Spain, Italy, Austria). Your nearest consulates will be in Los Angeles or San Francisco. I didn't check all the Schengen countries, but it's unlikely that the smaller ones will have a presence in Seattle if the major ones don't. Belgium has an honorary consul in Seattle, but visa matters are referred to Los Angeles.

There are no Schengen consulates/embassies in Washington state. The closest Schengen consulates to Seattle are located in Vancouver, BC. And at the very least the Dutch embassy does allow you to apply for a visa there as a Green Card holder living in Seattle.

So plan an itinerary where the majority of your stay is in a country which has a consulate in Vancouver and you can apply for a visa without traveling too far.