As far as I know the GAR was 'offically' integrated. But because the local post has to vote a person in, I am sure some posts would have not voted a Black veteran in. This would have been different from area to area. However, there certainly were some GAR posts with Black veterans.
Many faternal groups still take votes on membership and some local posts or chapters, are still basically segregated. I am in both the VFW and Sons of Union Veterans and both still have to vote a new member in. While both are 'officially' intigrated I would suspect some camps/posts would have a member or two who would not vote to let in some groups of people.