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The Federal Government does not ordinarily create or maintain birth, death, marriage or divorce records. Such records are made and kept by state and local governments rather than the National Archives.
Events before July 1, 1905: For birth, death, and marriage records dating from before July 1, 1905, you must contact the county recorder in the county where the event took place since records are not available at the state level. In addition, many birth, death, and marriage records dating before 1905 have been microfilmed by the Genealogical Society of Utah and should be available through your local LDS Family History Center. The California State Archives maintains only pre–1905 vital statistics from a few counties.
Also, please try using the genealogical websites Ancestry and/or FamilySearch as they are very popular platforms for searching for birth records. Please keep in mind that Ancestry is a subscription-based database, but it is available for free public use at all National Archives facilities and many public libraries. FamilySearch is free but you have to create an account with them. Similar to Ancestry, many libraries across the county offer free usage. You may also wish to read this helpful page on California Vital Records by FamilySearch.
In regards to documents that would indicate their presence there, there could be several. At the Federal level I would recommend that you first try to find your relatives in Census Records. As with birth records, you can also search Ancestry and/or FamilySearch as they both contain user-friendly search platforms for Census Records, as well as several other types of records that might interest you. At the State level, you could research land records, equity records, death or marriage records, or tax records. I would suggest that you reach out to the California State Archives for more help in researching your relatives at the State level.
We hope this information is helpful and best of luck with your research!
Civil birth records were not kept that early in most of the US. The exception were New England colonies and states kept town birth records very early.
This is a book with baptismal records for San Juan Capistrano starting in 1854. https://www.familysearch.org/library/books/records/item/63492-baptisms-1854-through-1910-mission-san-juan-capistrano-cal…
Both Ancestry and FamilySearch have a collection [California, U.S., Select Births and Christenings, 1812-1988] that cover that period, but my quick search showed very few for exactly San Juan Capistrano...
A better bet would be to search using Orange County instead
Here's the FamilySearch wiki which may be helpful