I need a culture/sensitivity test done on my EYELID,  but am not sure
where to find a good medical lab in Western Washington state that has
a lot of experience doing culture/sensitivity testing, especially for
the EYELID.  I understand a culture/sens test just for the eye is
quite common.

What information about this might be available on the Intenet (or by
phoning), and where might it be found?

It should  be easy for your doctor or whoever made the recommendation
to arrange it. If you haven't seen an eye doctor (opthamologist), that
person also would easily take care of it. In general, cultures are
taken by the requesting doctor, not the lab; the specimen is sent to
the lab for analysis. If this is coming from a self-diagnosis, I'd
strongly suggest going through your doctor instead.  Western
Washington is crawling with doctors. They all have easy access to
labs.

No, this is not a self-diagnosis.   There was a culture done by a lab
in my home state but they were not equipped to do sensitivity testing.
 It should be unnecessary to go to a specialist to have a
culture/sensitivity test done, particularly as it is typical taken by
a lab technician in the lab itself.