It depends on your state. I’m going to give you the facts and the law.
But before I do, I am going to suggest that you do not take anybody who has a child without a car seat. You can never live with yourself if a child dies in your vehicle even if the accident isn’t your fault.
In Most states, like the state of Florida, as a ride for hire. Meaning you are a taxi, TNC, or even a limo driver, it is at your discretion to give a ride but you are not required to provide a car seat for the child. It is the responsibility of the adult who was paying for the ride to provide a car seat for the child. If there is an accident that person is responsible for that child.
Some people suggest providing a car seat. I for one do not because if something does happen and the child is injured in the car seat that you provided you could be held liable