Does Uber ever carry car seats for children?

Yes, you may need to request it.

Yes, there is a relatively new program at Uber that offers car seats for children.  It is currently only available in certain cities, and there is a surcharge ($10 in NYC at the time of this writing).

More info:
https://www.uber.com/info/nyc-car-seat/
https://help.uber.com/h/3abcbae1-132b-42a9-8277-0dab00fa3879

You may also want to be aware that you are allowed to bring your own car seat as well.  Some drivers may get frustrated by the process and time it takes to install one, so you may want to warn the drivers immediately upon making a request.

You may also want to be aware that car seat laws with taxis and livery services are looser than they are for personal vehicles.  The topic is covered in the link below.  Unfortunately, the laws are currently not clearly defined for rideshare services where the drivers (say for UberX) may not have taxi or livery licenses.

Taking Taxis with Infacts, Toddlers & Young Children

Good question, but it's a tough one!

ippei 1202

Uber Driver, CEO of TaxiFareFinder and Unleashed, LLC