I wish you all the best in your search, because the Orphanage is/was part of the Catholic Diocese records are not available, when I bring it up at historical societies everyone rolls their eyes. My grandmother may have been a short term resident, she was adopted and there is no record of her from 1906 after her father died till 1910 listed as adopted daughter on a census.
I have a word document of information I have gathered but no records of any residents/children of orphanage. I will continue to add to my document (unable to attach as it is a docx file andn ot what Ancestry accepts).
St. Joseph Orphanage (1877 - 1980)
Send me your email if you would like, I am hoping to find something on a census, found a postcard of someone asking for a list of residents, do not have info handy right now.