The Arizona Department of Health Services has placed digital copies of their original records online. This collection includes birth records from 1865 through 1938.
The early birth records consist of delayed birth records created in the 1930s and/or affidavits. In some instances the affidavits are connected to the original entry for a person, showing under their name as “Document #2” or “Document #3”, but a few appear in the index as a separate record, so do be sure to check all entries for your ancestor’s name. There are also a number of records filed under the wrong date (one I found was indexed as 1879, but was actually a birth in 1897) or with an incomplete date, so even if the date appears to be wildly inaccurate for your ancestor you should check the actual record in case it was simply an indexing error.