I checked the 1982 book, "Adoption Agencies, Orphanages and Maternity Homes : An Historical Directory." Page 405 lists several orphanages in Wheeling that are Roman Catholic and old enough to have been in existence in 1909-10. The book is mostly for post-1923 listings, to serve living adoptees.
The "Official Catholic Directory" has been published by P.J. Kenedy for many years. During the early 1900s, its title was "The Catholic Directory, Almanac and Clergy List, Containing Complete Reports of All Dioceses . . ."
However, it seems to me that the federal census of 1910 would be the best source of information. Every institution and every inhabitant should be listed. Some people post queries saying it's difficult to find the right Enumeration District for an online census. If that is true, you could seek a volunteer with census experience on a message board.
There are over 2,000 queries on the board for Ohio County, WV, which would embrace Wheeling :http://boards.ancestry.com/mbexec?htx=board&r=an&p=l...
Sometimes the old records of orphanages wind up in the custody of historical societies and state libraries. In the case of a Roman Catholic orphanage, I suppose inquiries could be directed to the diocese or motherhouse of an order of nuns. The Official Catholic Directory (latest year) would have addresses.