I understand they 1915 NYS census for Mt. Loreto lists names of all the inmates, as they were called.
The home was owned by the Archdioces of New York, so they may have records or be able to direct you to them.
"The area that encompasses the Mount Loretto Unique Area was purchased from the Archdiocese of New York by the New York State Department of Environmental Conservation in 1999. Mount Loretto was once the largest orphanage and the largest farm in New York State, originally started by an Irish Catholic priest, Father John Christopher Drumgoole, founder of the Mission of the Immaculate Virgin in Manhattan, an organization to help poor and disposed children. He purchased the Staten Island land for the orphanage in the 1890's and named it Mount Loretto in honor of one of the nuns that worked with him at the mission.
"The property contained various buildings during its history as an orphanage including St. Elisabeth's girls dormitory a large Victorian Era building which was destroyed by arson during the time title to the land was being transferred to the state."