Is it permissible for a mother to travel to perform Umrah accompanied by her adopted daughter, the daughter's husband, and their children?

Adoption does not establish legal lineage or Mahram-relatedness, unless there is fosterage that makes them Mahram. Consequently, the husband of your adopted paternal cousin's daughter is not a Mahram to your mother, and similarly, his adult sons are not Mahrams to her. Therefore, it is not permissible for her to travel with them in this situation.