Is it permissible for the mother to slaughter the animal she vowed for her daughter-in-law’s pregnancy and delivery, and the son slaughters another animal for the Aqiqah of the newborn (male or female), thus fulfilling the mother's vow and performing the Aqiqah for the newborn?

Fulfilling a vow (nadhr) is a religious obligation. It is not permissible to combine the sacrifice for a vow and the aqiqah in a single animal, because each of them is an independent act of worship with its own specific reason, so one does not substitute for the other. The animal sacrificed for the vow only fulfills the vow, and the aqiqah requires a separate sacrifice. If the mother slaughters a sheep for her vow and the father slaughters a sheep for the aqiqah on behalf of the newborn, that is sufficient. It is more complete to slaughter two sheep for a male, though one sheep is also sufficient for a male.