I am from Racine and know a lot about the history there. St. Luke's is a hospital, but to my knowledge did not at all adopt babies out. There was however the Taylor Home orphanage - if you can prove direct descent from your grandmother, by your mothers birth certificate and yours, linking you all you may be able to access orphanage records if they exist.
As far as my research goes I know that you cannot access these types of files legally if you are not a direct ancestor (i.e. a niece). I would also just go to the Court House in Racine and ask if they have her birth certificate on file - if she is since deceased and you can prove it with a death certificate, you should have no problem.
You may also need the death certificate to access her personal files if any exist form the orphanage. Try the Court House, Library, and Heritage Museum - they are all in very close proximity and may help.