According to a book titled, "Search and Research," by Noel C. Stevenson, Minnesota conducted a state census in 1895. The Minnesota State Archives, in St. Paul, had custody of all the state censuses at the time the book was published (1973).
I am not an expert in Minnesota genealogy, so I don't know if these censuses were indexed by name. If you are willing to search all of St. Paul there is a possibility you will find the entry for your grandmother and the name of her orphanage. Her entry may include something if her parent was of foreign birth---I don't know if the country of his birth is given.
With the name of the orphanage, you can post a new query seeking the records thereof on this message board (at Ancestry.com) and the message board for Ramsey County. You could also do a Google search for the name of the orphanage to see if someone started a webpage about it. You could also seek its records in local historical societies and a state library, if any.