If the cab driver is not a parent or legal guardian too and they verify that the child or ward is accompanied by a parent or legal guardian as passenger, no and there are other exceptions.
Quoting Vehicle Code § 27360,
(a) Except as provided in Section 27363 a parent, legal guardian, or driver shall not transport on a highway in a motor vehicle, as defined in paragraph (1) of subdivision ( ) (c) of Section 27315, a child or ward who is under eight years of age, without properly securing that child in a rear seat in an appropriate child passenger restraint system meeting applicable federal motor vehicle safety standards.
(b) Subdivision (a) does not apply to a driver if the parent or legal guardian of the child is also present in the motor vehicle and is not the driver.
Note that (b) is an important clause in some cases.
Further exemptions are provided for in Vehicle Code § 27363 do not include anything general related to transport to or from airports; most relate only to the regulations that state that children must be in a rear seat. Here are the main ones:
(a) The court may exempt from the requirements of this article any class of child by age, weight, or size if it is determined that the use of a child passenger restraint system would be impractical by reason of physical unfitness, medical condition, or size. The court may require satisfactory proof of the child’s physical unfitness, medical condition, or size and that an appropriate special needs child passenger restraint system is not available.
(b) In case of a life-threatening emergency, or when a child is being transported in an authorized emergency vehicle, if there is no child passenger restraint system available, a child may be transported without the use of that system, but the child shall be secured by a seatbelt.
(c) A child weighing more than 40 pounds may be transported in the backseat of a vehicle while wearing only a lap safety belt when the backseat of the vehicle is not equipped with a combination lap and shoulder safety belt.
(d) Notwithstanding Section 27360, a child or ward under eight years of age who is four feet nine inches in height or taller may be properly restrained by a safety belt, as defined in paragraph (2) of subdivision (d) of Section 27315, rather than by a child passenger restraint system.
(e) Notwithstanding Section 27360, a child or ward under eight years of age may ride properly secured in an appropriate child passenger restraint system meeting applicable federal motor vehicle safety standards in the front seat of a motor vehicle under any of the following circumstances:
(1) There is no rear seat.
(2) The rear seats are side-facing jump seats.
(3) The rear seats are rear-facing seats.
(4) The child passenger restraint system cannot be installed properly in the rear seat.
(5) All rear seats are already occupied by children seven years of age or under.
(f) Notwithstanding subdivision (e), a child shall not be transported in a rear-facing child passenger restraint system in the front seat of a motor vehicle that is equipped with an active frontal passenger airbag. (6) Medical reasons necessitate that the child or ward not ride in the rear seat. The court may require satisfactory proof of the child’s medical condition.