A. E. Jordan
Here's a sort of primer on naturalization records for New York City.
Naturalizations in the New York City area happened either in the Federal or State courts up until the 1920s. The Federal courts were the Eastern District which is in Brooklyn and the Southern district which is in Manhattan. There's also a Northern district north of the City.
The State courts are named the Supreme Court and there was a location in each of the boroughs.
Here's where it gets challenging. There was no residency requirement meaning you could live in Brooklyn but work in Manhattan and so you decided it was easier to go to the Manhattan court to file your papers or any combination of courts. Also there was no rule that I am aware of the governed going to the State vs. Federal court.
And to make it even more challenging in some rare instances the person started the process in one court and then moved and decided it as easier to finish the process in another court closed to where they now lived. To use a 21st century term, the naturalization was portable. As a matter of fact I was jut helping someone last week who had the declaration number for their ancestor and when we pulled up the online record attached to the declaration is a letter saying a duplicate had been made and when you look at the two you see he was in Manhattan when he started the process but was in Brooklyn when he was requesting the copy which leads to he assumption he finished it in the Brooklyn courts.
Ancestry, FamilySearch, Fold3, the Italian and German gen pages all have indexing to parts of the naturalization process. I personally like the Italian/German gen index but you have to search all of them sometimes.
The 1925 NY State Census asked people what court they have naturalized in so you often find little notes on the names but of course it is not always accurate.
Also on the later passenger lists you often find a notation with the date and a number that is a clue to when and where the person naturalized. I had one several years ago and no one seemed capable of deciphering it including NARA. Marion Smith ended up helping me and it turned out the man had done his naturalization in Westchester County.
FamilySearch had a lot more of the naturalization papers from NYC than what they have indexed. The images are loaded but you have to manually search the files using the petition number. An easy way to see what exists is go to the National Archives NYC website and click on their naturalization records. It will bring up a table called "Our Holdings" and on the far right side is a link that generally jumps to the files on FamilySearch. Thee good news --- although it is years away -- NARA is funding a project to digitize more of the local court records from the State courts.
Most of the local courts will do mail order copies (once of course the current public health crisis is lifted) if you can not find the record online at one of the databases mentioned above. In Manhattan the original books are sitting on the shelves of the old court record room (7th floor in the same building as the Municipal Archives) and they let you pull them out and take photos on your phone. Hidden away on the 8th floor and the supervisor has to get it are the original ledger index books and he also has the originals of cases where people did a declaration but never came back to do a petition. Those incomplete files are in separate books from the ones where the person completed the process.
NARA also will do copies but normally when you go to NARA in NYC (they are closed due to the health crisis) they will first point you to the computer resources I am mentioning here. NARA NYC has the original paper files for the records after the digital indexes so if it is post-World War II for example you need to get it from them on paper. If you go in person they only pull records hourly a few times a day but the charge is only 25 cents a page but has to be paid by credit card .... they stopped handling cash years ago.