Yes, it is possible to adopt twins. According to the Independent Adoption Center, an open adoption agency in the United States, adopting two children at the same time is possible if a family talks to a birth mother who has twins.
The Independent Adoption Center reports that sometimes adoptive parents want to adopt two children from two different birth mothers at the same time. However, this usually results in the birth mothers backing out of the adoption and the adoptive parents becoming frustrated. Therefore, the adoption center has created the policy that adoptive parents are only allowed to adopt children from one birth mother at a time.