Both parents are most commonly required to show up when doing a child’s US passport application. While there are ways around both parents being required to show up to get a child’s US passport, it will risk slowing things down and could mess up things.
A child’s rather recently expired US passport along with both parents presence with their own valid passports or other acceptable ID works. Having a certified copy of the birth certificate wouldn’t hurt and generally shouldn’t be needed ( but that may vary based on the circumstances relevant to the child’s expired US passport). More importantly, get the photo done correctly.
With in-person morning applications at the passport agency offices, they can do a same-day production for in-person pick-up on the same day or next day. Even now. Make sure to highlight imminent travel schedule as there is some internal prioritization that makes a difference in the turn time.