Our company, headquartered in Washington State, has a vocational rehabilitation counselor in our employ who works from his home in Delaware. Is he required to post labor law posters in his home office?

Unless there is more than the employee working there, it would be silly to post something like that in a home. Instead, to completely protect you, I would say that the employee should make sure that he/she received the poster and has one on file, should questions be asked.