Based on the time frame you are researching, and the Oberlin location, my guess would be that you are looking for The County Home For Children, also known as the Lorain County Children's Home. In later years I believe it was known as Green Acres Home, although I am not 100% sure of that. The home was located on the eastern edge of Oberlin.
A 1916 History of Lorain County, devotes about four full pages to the home, and includes a full page picture of the Home. The text provides interesting detail on the buildings and accommodations for the children, as well as a list of conditions under which children would be placed with families for adoption. I have a reprint of this history, and, as with most reprints, the pictures tend to leave something to be desired. However, the local public library, I believe, has an original release of the history. If you wish, the next time I am at the library I will be happy to copy the material and mail the copies to you.
The children's Home was completed and started operation in 1899, and contained about 15 acres of land. I don't know where you would go to find a list of children who passed through the home, but as a thought, I wonder if the 1910 census would list the names of the children in residence at the home. If you are real lucky, you may find the names of the children you are seeking, assuming they were living at the Home at the time of the census.
If you would like me to follow up the information on the Home, you can e-mail me directly at