Can I earn money by dog sitting in addition to my existing day job, while on H1B visa?

You visa allows you to work in the USA for the company that sponsored you so I highly doubt that you can also become a sitter on the Rover platform. But to double check you should call Rover support @ [Edit: Rover’s contact options have changed. Visit the Rover Help Center at https://support.rover.com/ to find the phone number, help articles, or chat with the team]