My Grandmother immigrated to the US from England in 1911. She appears as Sophie Charney in the 1920 US census, but in 1924 she appears as Shirley Charney in the New York State marriage record to my grandfather, Herman Emelock. I'm trying to find the record of her name change which would have occurred between 1920 and 1924.
Try looking in local courts in the areas where she was living. They may not be in a collection which is titled as such -- for one example, see the FamilySearch Research Wiki article New York Names, Personal which cites a collection Miscellaneous Filed Papers Index, 1812–1934 for New York County.
The period you're looking at is post-Consolidation (see: Wikipedia, City of Greater New York), so be aware that records for New York City are often kept separately from the rest of the counties in New York State.
For an overview of name changes for immigrants, see the New York State Archives' research guide Naturalization and Related Records: Records of Name Changes. Their guide says:
The State Archives holds the lists of name changes filed in the Secretary of State's Office for the period 1899-1940 (series B0070; not indexed).
Another possibility for a formal record of a name change might be in naturalization records. Since she married in 1924, her naturalization status would not depend on her husband's status (it is after September 22, 1922, when Congress passed the Married Women's Act, also known as the Cable Act). Petitions for Naturalization sometimes say what name an immigrant had on the passenger list as well as the name that they were going by at the time of the petition. The US National Archives' online guide Clues in Census Records, 1850 - 1930 shows how you can find clues to her citizenship status in different census years (including 1940, despite the title of the page).
While Naturalization records are Federal records, they need not be in Federal courts. Petitions can be found in local courts well into the 20th century. See Directory of courts having jurisdiction in naturalization proceedings (1932).
For an overview of US Naturalization laws pertaining to women, see Marian L. Smith's article from Prologue Magazine, Summer 1998, Vol. 30, No. 2 | Genealogy Notes “Any woman who is now or may hereafter be married . . .” Women and Naturalization, ca. 1802–1940.
For current rules on name changes, see Name Change Basics at the website of the New York State Unified Court System (NYCourts.gov). The section Where to Go says:
A name change request can be made in the County Court or Supreme Court of the county where you live. If you live in New York City, you can go to the New York City Civil Court or the Supreme Court in the county where you live. The Civil Court costs less money than the Supreme Court.
You could start with the County Courts and the Supreme Courts, if you want to assume the records might be found in those courts as they are now. But if you want to find the law for name changes at the time you're considering, it is possible to find the current statute and then walk backwards using the references in the statutes to the historical period in question. See NY State Codes & Statutes at NYCourts.gov.