When you purchased a firearm, your name, date of birth and home address was recorded in the Automated Firearm System (AFS) along with the firearm make, model and serial number. Starting July 1st, 2019, that information will be used to verify you are qualified to purchase ammunition. California Penal Code 30370 states all ammunition purchases will require a DOJ eligibility check. The check will cost $1, and also verifies there isn't anything on your record like a felony conviction that would prevent you from purchasing ammunition. If your name or address does not match what's in AFS, or you have never purchased a firearm, you can pay a one-time $19 elegibility check fee.
The CA DOJ should update the California Firearms Application Reporting System (CFARS) so you can register firearms or update personal information like a change of address. The CFARS website is cfars.doj.ca.gov.
You can contact the CA DOJ to see if they have your correct information on file.
When you purchase ammunition, they will record the date of sale, your name, address, telephone number, date of birth, driver's license number, what you purchased and how much. That information will be sent to the CA DOJ. There isn't anything the CA DOJ will do with that information necessarily, but chances are they will store that information unless required to delete it by law.