Here is what little I know about the St Joseph's Orphanage or as it was called then; St Joseph's Orphan Aslyum. It burnt down and most of whatever records burned as well, except for the original register of the children who were delivered there. The information in this register is limited to whatever the person who brought the children could give them at that time. This register is kept at the Tucson Catholic Church.
You can contact Diocese of Tucson Archives for
Archives & Library
8800 E. 22nd Street
Tucson, AZ 85710
W-TH 9am - 2pm
I hope this helps you in your search.