Your note said that “No naturalization records have been found for either Giovanni, Carolina, or Giuseppe.” http://www.digitalarchives.wa.gov/Home has this information for Giuseppe Varriano LaFazia:

Washington State Archives • Digital Archives
Kitsap County Clerk, Naturalization Records, 1906-1910, 1929-1955
Name: Giuseppe Varriano LaFazia
Alias Name: LaFazia, Joseph
Naturalization Id: 597
County: Kitsap
Document Type: Petition for Naturalization
Volume: 9
Document Number: 1237
Birthdate: 8/22/1895
Country Of Origin: Vinchiaturo, Province Campobasso, Italy
Entry To Us Date: 6/23/1908
Date Of Document: 2/11/1943
Spouse Name: Rose
Childrens Names: Caroline Marie Rice; Joseph Francis; Helen Thelma Henry; Rose Elizabeth; John Albert; Betty Jean; Francis Rex
Volunteer Notes: Emigrated from Naples, Italy. Petition continued four times (each time for further study); petition ultimately denied (List #81).
Other Document: