No orphanage in Wheatland appears in this directory :
UNITED STATES. Bureau of the Census.
"Children under Institutional Care, 1923: Statistics of Dependent, Neglected, and Delinquent Children in Institutions and under the Supervision of Other Agencies for the Care of Children . . ." Washington, D.C., 1927. 381 p.
In this special census of 1923, personal information about individuals was only taken from delinquents and inmates of homes for unwed mothers. However, this book (which I saw in the New York Public Library) does not list individuals.
I also checked "Adoption Agencies, Orphanages and Maternity Homes: An Historical Directory" (published in 1982). No entry for Wheatland.
If you prefer to determine the name and locale of the orphanage by searching for the surname FITCH in the federal census of 1920, Ancestry.com has a message board for genealogists interested in Platte County, WY :http://boards.ancestry.com/mbexec?htx=board&r=an&p=l...
Try to find a genealogist there who has a subscription to Ancestry.com and access to the index of names for the federal census of 1920, which would lead to the number of an enumeration district (I guess). Then ask someone to look up the census and give you the age and whatever details there are for Elenor FITCH, including the name and locale of the orphanage.
Wyoming conducted a census of its own in 1925, but it cannot be found, according to a genealogy manual published in 1973 : "Search and Research," by Noel C. Stevenson. I am not an expert on Wyoming, so I cannot vouch for Stevenson.
To order a death certificate for Elenor, obtain the address of the vital records office from :http://www.cdc.gov/nchs/howto/w2w/wyoming.htm