Answer:
Yes, Uber and Lyft drivers can demand that the at-fault driver's insurance company pay them for lost income, also referred to as a "downtime claim."
You will have to provide documentation of your earnings from Uber or Lyft. The more documentation you provide, the better.
The other driver's insurance carrier will try to fight your claim, arguing that you could have mitigated your losses. But Uber and Lyft will not let you drive a rental car on their platform, therefore it is impossible for you to mitigate your losses.
The insurance carrier will also attempt to lessen what they owe you by claiming that you delayed the process, and they are not on the hook for any downtime loses attributable to your delay. But as long as you moved swiftly to get your car repaired or the total loss paid, they are on the hook for your lost income from Uber or Lyft.