I quit driving for Uber in 2017. At that time, they had discontinued accepting drivers into the car seat program for at least a year. They continued to offer car seats in the app, but it was limited to drivers already enrolled in the program. If they have reopened the program to new drivers, I am not aware of it, but it's possible.
Also, over the years, there have been a number of posts on this board by parents saying an Uber with a car seat was not available when they needed one.
Backup plans might be a plan to use Disney Transport. Also, some Mears taxi drivers have been known to carry a car seat, but I don't think there's a way to request one. That was before Mears was sold to outside investors, so I also don't know whether that has changed.
Just for fun, I just did a test request of a Uber with car seat from CR to Pop. Here's the result: (UberXL with car seat was not offered as an option.)
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