My great grand parents both came from Italy (separate times). Around 1911 and 1921. My grand mother was born in the US. My grandmother wasn't a US citizen. We are looking to see if my grandfather was or not when my grandmother was born. If he wasn't a US citizen when my grandmother was born, would Italian citizenship be passed from my grandmother (born 1929), to my mother (born 1951 in US), then to me?
If your great grandfather was still an Italian citizen when your grandmother was born in the US, she obtained Italian citizenship from him. She would have passed that citizenship to your mother, who would have passed it to you. The naturalization date is critical here.
I never really had to do a search, so I can't provide much guidance. You might want to try beginning a new thread with your great grandfather's name and tentative dates. Someone here may be able to help.
I'm confused why you say your grandmother was born in the US, but was not a US citizen? Under American law, anyone born on US soil is a entitled to American citizenship, regardless of the citizenship of the parents.
If your great grandparents lived in New York State, you would contact the County Clerk in the county they resided to request a search of naturalization records in the state Supreme Court. You will also need to contact the National Archives and the Dept. of Homeland Security to search their naturalization records.
But before you do that, you may want to check all the census records available that lists if a person was a citizen or naturalized (although sometimes the clerk wrote down the wrong information). In your great grandparents were alive during the 1940s, you can check to see if their is an Alien Registration form which is proof if they remained aliens.