Shot records are also given to day care and schools. As you can prove legal custody, perhaps you can contact anyplace (camps, schools, day care) your child attended and simply get a copy of at least the immunization records from them?
Having lived with the fact that I will never have a full and complete medical history of my child's first two years, I understand your frustration. However, It IS possible to determine if the child has acquired immunities (from shots or exposure) via a test when blood is drawn. Many of us used such a test on our orphanage kids when they came here, as we were uncertain of both the reliability of the foriegn records and the quality and storage of any immunizations they did recieve. Mine was reimunized for a couple things to be certain.