Orphanages were enumerated in the 1880 census.
I tried "orphanage" as keyword at 1880 United States Federal Census and got 40 results, but none in Arkansas. Most results matched "orphanage" in the occupation field, but linked census images had many child boarders or students. Creative searching using related keywords may find more results.
In later censuses, institutions (hospitals, orphanages) often had their own ED (enumeration district), but you would have to be able to pinpoint a county or township.
Don't limit yourself to a single state: in one of my related families, 3 Ohio-born children were placed in a church-run Michigan orphanage by their widowed father.