According to tradition going back to the second century (documented by renowned Scripture scholar Cornelius a Lapide), Christ baptized the Blessed Virgin Mary.
It is true that Blessed Mary was without original sin, and full of grace. However, she had yet received the baptismal character, which is an indelible feature of the baptized. Also, the sacrament of baptism linked her even more closely with the Church of her beloved Son.
If she humbly submitted to Old Law precepts out of humility (e.g. the purification after birth), then she no doubt would also have received the sacraments of the New Law. Moreover, Mary wanted to be like Christ her Savior in all ways. Christ didn't need baptism but He received baptism. No doubt, she desire the same. Mary didn't need to die, but she desired to die in order to be perfectly conformed to Christ. This is a feature of all the ancient Dormition accounts.
Did the Immaculate Mary need to be baptized. No, of course not.
Did the Immaculate Mary want to be baptized. Yes, of course.
ad Jesum per Mariam,