these petitions are indeed available on Family Search - just not indexed nor are they under "Records"
Search in "Catalog"
then under "Place" type in "United States"
scroll down to the 4th page where you will find United States - Naturalization and Citizenship"
under that wil be a drop down of many different curt records - NYC will have Eastern and Southern Districts if you search in the scroll down
- also many other localities.
Find the court based on your indexcard listing
the armed with your petition number from the index you should be able to locate your record - it will involve trial and error but not too hard
Hope this helps