I highly doubt they'll allow him to take leave. My husband had 3 family members pass away within the same week and they let him come home for 2 days for the funerals only because they were worried about his mental health. They wouldn't have let him come home during AIT if it hadn't been for that 3rd passing.
They may let him call or skype during labor. It will just depend on a number of factors. Some soldiers end up getting to skype for the entire birth, where as others may only get one phone call. Whatever happens, just remember he can't do whatever he wants.
Depending on the phase he's at during AIT, he may be allowed to have his own laptop/cellphone after they're done for the day and before bed. During the 2nd phase they get a little more freedom. He's not guaranteed anything though even if he phases. If a soldier fails a test they can be pushed back to phase one and lose any privileges that may have been given. That may be some motivation to work extra hard and pass everything thrown at him. Good luck