MInnie Van hours are 6:30 am to 12:30 am according to the Disney website. But I'm not saying that the Disney website is always 100% accurate. The link has the phone number for requesting an accessible Minnie Van. Maybe if you call they can give you accurate hours for a regular Minnie Van. https://disneyworld.disney.go.com/guest-services/minnie-van-service/
Uber also has some vehicles with ONE child seat, but that is subject to availability of course. You'll be able to tell on your app right away whether any are available. Lyft does not have child seat vehicles AFIK.
Maybe on a holiday
I just rode in one yesterday
The driver said they start work shift at 6:00, rides at 6:30
Came up while we were asking about her job and she talked about her recent shift change
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