Neither Uber nor Lyft have any legal claim to a driver’s vehicle, which by the Federal government is recognized as private property. So, in other words the companies can’t dictate whether drivers have a camera or not.
Now, in terms of the law it really depends on the jurisdiction. When it comes to recording people and their conversations, the laws are complex, and vary by State and potentially by the city. In most cases, your driver will need to at least inform you that you are being recorded. Now, in some cases they don’t need permission at all assuming it is on their property or inside a private vehicle. In others they need explicit permission, or a notification of the camera needs to be displayed inside and outside the vehicle.
Now yes, you can ask. But the chances of a driver turning of the camera for you is highly unlikely you are better off just canceling the ride and taking the cancellation fee. Just know most drivers have dash cams, and the ones who don’t we highly recommend they get one. For drivers have dash cams to protect themselves and you.