I just confirmed my first grooming appointment online including authorizing payment but I have not received confirmation of my chosen date and time.
If you are asking this question of Rover, its customer service people do not come here. Try calling [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] if you have questions about a specific booking or the person you actually booked because if you didn't receive anything, then he/she didn't accept the booking.