There are 4 rescues in Washington, but none are in Seattle. I don’t know how close any of these are to Seattle, but maybe one can help.
Rivers Wish Animal Sanctuary, Spokane, Washington
Rabbit Haven, Gig Harbor, Washington
Rabbit Meadows Sanctuary – Best Little Rabbit, Rodent & Ferret House, Poulsbo, Washington
Bunanza Rabbit Rescue Ranch and Adoption Center, Lynden, Washington
Country Road Rabbit Rescue, Gig Harbor, Washington
I volunteer at a rabbit sanctuary and these are the instructions we give about catching a rabbit:
There are 2 methods of catching a rabbit. The best way is the people method, the other way is using a humane trap. Sometimes a single person can get a rabbit, other times, it takes a small group. If you can get the rabbit used to you, by feeding it some treats, such as apple slices, banana slices, other fruits, cilantro, parsley, etc. Then if you are close enough and can grab it, immediately put it in a carrier and secure it. Rabbits do not like to be held and usually struggle, causing them to be dropped and possibly injured, then they will likely run away and you won’t get another chance. That’s why you need the carrier at hand and you need to get a firm grasp on them, without hurting them, of course. It may be with a few more encounters with the rabbit, that he lets you get close enough to get him. If he gets weaker due to malnutrition or possibly gets injured, that also might make him easier to catch.<br aria-hidden=”true” /><br aria-hidden=”true” />
If helpers are needed, you get a few people and surround the rabbit, never chasing it. It’s good to corral it up against a fence or building or in a garage, then cut off its means of escape. Sections of an x-pen are good to carry around to cut off exits and keep it contained until you can get it in the carrier. You could even throw a blanket or quilt over him to slow him down and hopefully get him. Here is one article that gives some pointers:
https://www.columbusrabbit.org/tips-for-catching-a-stray.html
The other method for catching a stray rabbit is to use a Hav-A-Hart trap, baited with apple, banana, pear. Some Humane Societies will loan out the traps. The trap has to be closely monitored. If the rabbit does go in, it can hurt itself from fear or a predator can come upon it. So you would need to get the rabbit out of the trap and into a carrier as soon as possible. If you transfer the rabbit from the trap to a carrier inside a house or building, that would be safer, with less chance of the rabbit getting away. You may also catch possums, raccoons or other animals. That’s why we prefer the people method.