Dear Ms. Westbrook,
Thank you for posting your request on History Hub!
The National Archives and Records Administration (NARA) is the depository of the permanently valuable non-current records of the United States Federal Government. Under Italian law, records of births, marriages, and deaths are maintained by the Registrar of Vital Statistics (Ufficio dello Stato Civile) in the city (comune or municipio) where the event occurred. There is no central, regional, or provincial office established which keeps such records. Therefore, we suggest that you contact the Stato Civile - Città di Lucca for guidance on how to obtain the birth records that you seek. Please review the FamilySearch Research wiki for Lucca, Tuscany, Italy Genealogy and the U.S. Embassy in Italy webpage on Obtaining Vital Records for more information.
Prior to September 27, 1906, any "court of record" (municipal, county, state, or Federal) could grant United States citizenship. Often petitioners went to the court most geographically convenient for them. As a general rule, the National Archives does not have naturalization records created in state or local courts. However, a few indexes and records have been donated to the National Archives from counties, states, and local courts. We suggest that you contact the National Archives facility serving the state in which the petitioner resided to determine if records from lower courts are available. If the records are not available, we suggest that you request a search for the naturalization records from the State Archives or County Historical Societies.
For a copy of your great grandfather’s death certificate, we suggest that you contact the Office of Vital Records for the state in which they resided.
We hope this is helpful. Best of luck with your family research!