The 1880 Enumerator instructions, which can be viewed at ipums.org, say:
Wherever an institution is to be enumerated, as a hospital, an asylum,
an alms-house, a jail, or a penitentiary, the enumerator will leave
three lines blank, and enter the name of the institution (as "St.
Mary's Hospital," "Protestant Orphan Asylum," "Insane Asylum," "City
jail," etc.) above the names and the inmates. All persons having their
"usual place of abode" in such institutions, whether officers,
attendants, beneficiaries, or persons in confinement, should then be
entered consecutively on the schedules, and at the close of the list
another space of three lines should be left blank.
The FamilySearch Wiki article United States Records of Childrens Homes and Orphanages (National Institute) says:
By the 1880s, most states had enacted laws requiring children to be
cared for in facilities separate from adults. It is at this time you
will see the creation of government-run county children’s homes.
The term ‘orphanage’ is often a misnomer. These institutions often
took in not only parentless children, but also children whose parents
were unable or unwilling to provide for them. Such children could be
placed there either by the choice of their parent(s) or by the courts.
Let's assume for the moment that the family story is true, and you won't be able to find any records from the orphanage itself. What you might be able to find are records that mention the orphanage by name, like the census, or there might be other records created by other agencies, like the local courts. The article in the Wiki article on Burned Counties Research (see the resources) was written for situations where the county courthouse has burned down, but the same principles apply to any record loss. Be sure to search all levels of jurisdiction (federal, state, county, and smaller localities) for records. Arkansas state statutes and those for surrounding states can tell you at what point children were placed in institutions separate from adults. Check online (Google Books, Hathi Trust, Internet Archive) for local histories that might mention the names of institutions operating at the time, then use that information to search the US Federal Census.