How does one go about getting Rover to paying me through my S-Corp if I'm already a registered Rover Sitter?

You are a Subchapter S? Fancy!

Call Rover Customer Support at [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] to get your Tax ID changed as well as how to get the payment instructions changed on Stripe.