In NYC, which I think was the pilot program, Uber provided car seats and instructed and checked out the drivers on installation. In Orlando, drivers are given a list of approved car seats (I think there were about 10) and have to buy one at their own expense. They then have to submit a picture of the label to prove it's current, and a picture of the seat installed in their vehicle. There was a required "brief training session," but I don't know exactly what that entailed.
Did you know you can start the Uber app and set your pickup location to anywhere just to check availability, and also get a fare estimate? I just did that for CSR, and it initially showed no car seats available, and then it went to 19 minutes estimated wait, so apparently the car seats are not plentiful.