Public information will depend upon when your mother was in an orphanage. Census records for the orphanages are only available up to 1930.
Now, I'm not an expert at this. But this is what I was told when I searched for my dad's records and those of his siblings for the Charleston Orphan House.
Records have a 75-year life term, meaning they are not available PUBLICLY until the records are 75 years old. If you can figure out which orphanage your mother was at and when, you can contact them and either request copies (usually for a fee) and have them sent to you. I also think that you can go in person with your mother's obituary or death certificate, your personal identification and request copies.
The city of Charleston has all the records from the Charleston Orphan House. You may want to contact the records manager there to see if your mother's name comes up. Here is her email address: WELSCHS@ci.charleston.sc.us
City of Charleston
Records Management Division
P.O. Box 304
Charleston, SC 29402-0304
This is what I know from my experience with the Charleston Orphan House. I don't know about any other institutions. Privacy laws have affected a lot of the information available.
Good luck. I hope this was somewhat helpful to you.