Naturalization records c. 1821-1954 are housed at the County Historian's Office. Researchers seeking proof of a women's citizenship or naturalization of a person who arrived in the U.S. as a minor child may find themselves stuck. Here's a quick overview:
Nearly all naturalization paperwork before 1922 was completed by immigrant men. While it was not illegal for a single (spinster) or widowed immigrant women to petition a court for citizenship, there were few incentives. In New York State before 1918, no woman could vote, few held property, and there were unappealing court fees associated with citizenship proceedings. Between 1855 and 1922, married alien women would have been very unlikely to have been granted citizenship individually from a husband.
Therefore, women became citizens automatically upon the naturalization of their immigrant husbands, or upon marriage to a native-born or naturalized man. Likewise, minor children born outside the U.S. would automatically become citizens when their father was naturalized. If this did not occur before a boy was of age, he could apply for himself. It would have been assumed that a foreign-born girl would achieve citizenship through marriage. Before 1906, immigrant wives and children's names were almost never recorded in the naturalization paperwork. After 1906, the required forms became more detailed.
In 1922, all women, married, single, or widowed, could complete naturalization paperwork independently.
To read more about this subject, check out the National Archives' page on women and naturalization.