You might try Mary’s Place, they have a shelter in White Center,
Here is their main office (Downtown) phone number:206.621.8474. I don’t know if you can directly drop off at the shelter, but they undoubtedly could tell you.
Thank you, miws! They have 2 drop off locations and the one in Belltown is right on the Rapid Ride C Line route. I appreciate your help and have a great day!