A Catholic directory published in 1922 shows House of Providence at 1420 W. Onondaga Street, Syracuse. It was then staffed by---or under the auspices of---the Daughters of Charity of St. Vincent de Paul, with motherhouse in Emmitsburg, MD.
In 1944, The "Official Catholic Directory" listed its name as St. Joseph's House of Providence and its address the same as in your query. The 1976 edition of the same directory shows House of Providence at the same address as your query---an address shared by Catholic Social Services. This edition does not mention the order of nuns, but the 1964 edition says Sisters of Social Service, which may or may not have a motherhouse in Buffalo, NY.
Researchers interested in Catholic institutions, seeking the location of old records, can inquire of the Diocese (the address of which should be in the latest edition of the "Official Catholic Directory") or the latest known order of nuns or an historical society in the area. One can also Google the name of the institution to see if someone has created a website devoted to it. When searching for an exact name consisting of more than one word at Google, use quotation marks around the search term : "House of Providence"
Children in the orphanage would be listed in the federal census of 1930, as well as earlier and later censuses. Ancestry.com says it created a name index to the 1930 census, available online to its subscribers. Experts who hang out at message boards devoted to individual counties sometimes volunteer to look up individuals in censuses.
There are also New York State censuses earlier than 1892, plus 1892, 1905, 1915, and 1925. These may be at the various county clerks and/or the New York State Library in Albany. Probably also at the Family History Library in Salt Lake City.