I am going to mention the "L" word, so stop reading this message if it would offend you. The "L" word is "Library," a place noone on the Internet wants to go.
But if someone went to a large library, he would do well to seek a book about institutions in the USA :
UNITED STATES. Census, Bureau of the. Benevolent Institutions, 1904. [Washington, D.C.: Govt. Print. Off.], 1905. 335 p. (Special Reports). A directory of 4,207 institutions (excluding 85), with their addresses and statistics: orphanages, hospitals, temporary homes for adults and children, etc. Includes a single alphabetical index to the institutions.
There may or may not be addresses for many of the entries in that publication and there certainly won't be the names of the inmates. But their names would be in the regular federal censuses of 1900, 1910, 1920, etc. Ancestry.com has all of them online for the benefit of its subscribers.
Ancestry.com also has message boards for the various counties in the various states. You can search for a county in the search window at the bottom of every page, where it says "Find a board." By reading the queries and responses on those boards, one might be able to identify a volunteer with experience in searching censuses.