All of the Rover services (boarding, walks, drop-ins, etc.) that are booked through the website are paid for once everything is confirmed (owner and sitter). Like Paige and Cari, I would not start a service unless the booking is confirmed (no pending owner response, etc), even if that meant meeting the owner at your door with a computer so they can finish the booking.
However, if you're asking about something extra - like needing to pick up dog food, that's tougher. Rover has no way to request the owner pay you through their site, so you have to ask the owner to please bring you cash or check, or pay electronically through Venmo, PayPal or Square Cash. I would make sure to document anything like this via Rover's messaging system, and send photos of the receipt . If an owner won't pay you, you can try calling Rover Support for help, but I have a feeling there isn't much they can do either. Unless it's a large amount of money, you may have to chalk it up to a learning experience and avoid that client in the future.