Dear Ms. Havey,
Thank you for posting your request on History Hub!
Because your great grandparents resided in New Jersey, we suggest that you contact the staff at the National Archives at New York via email at email@example.com to have them perform a search through the naturalization records they maintain from the Federal courts in that area. Please be sure to provide all known name variants in your request to allow for a successful search.
We also suggest that you review NARA’s website for tips on how to begin your genealogical research; and Census Resources for how to access census records that may are digitized in the National Archives Catalog or available on our partners’ websites as well as how to view them for free at a NARA facility. NARA has census schedules on microfilm dating from 1790 to 1930 but data from recent censuses are not available after 1940 due to the 72-year restriction on access to the Census. Most researchers find it helpful to begin with the 1940 Census and work backwards to locate people in earlier generations. There is a citizenship column on each census starting in 1900 that can provide clues to an individual’s citizenship status within the U.S. An “AL” indicates alien status, “PA” indicates that the naturalization process was started, and “NA” indicates that citizenship was obtained.
You may use Ancestry to access Census records and other digitized records. There may be a fee for using Ancestry. However, you may view these records online at one of NARA's facilities for free via a NARA PC. For the nearest NARA location, please consult our web page at https://www.archives.gov/locations/. Also, some of the images from Ancestry are available for free on Family Search at https://www.familysearch.org/en/.
We hope this is helpful. Best of luck with your family research!
[Some information provided by Elizabeth Burnes, Subject Matter Expert]
You could also check census documents up to 1940. It usually states whether they are naturalized or not. 1950 census will not be released until 2022.