Well this is awkward.
Aurelia: To become the strongest Huntress and to become a beacon of hope for Vale.
Airi: To prove her worth and silence her detractors. Her deepest desire would be to successfully succeed her adoptive father.
Avery: To become the headmaster of Atlas.
Sophea: To become Chief Practitioner of Vale's most renowned hospital (alternatively, a medical researcher), and to bring medical innovation to the Plains.
Mao: To be relieved of the expectations (that he feels are) weighing on his shoulders, and to relieve his mother of her inherited responsibilities.
Izumi: To get her parents back, and to return to the life she once had. It would've been to kill the criminals who destroyed her family, but she already checked that one off.
Yu: Being a multi-faced punk and playboy, his wish right now is simply to be happily married.