The Unnamed social worker is an unnamed woman who works at Springfield Orphanage.
When Bart came to the orphanage, he asked a worker if he could adopt a little brother. She sent him away because he is too young, but she didn't notice that an orphan named Charlie followed Bart home.
When Charlie had to return to the orphanage, Bart and Homer later came to visit him. However, the worked told them that Charlie had been already adopted.