Owner hasn't responded and I'm sure the puppy has a UTI and is very uncomfortable. I want to take her to her listed vet tomorrow, but the vet couldn't find her account. If I don't hear back from her before I take her to the vet, how do I get reimbursed for the visit?
You don't. Call Rover 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] for assistance.
I would not take a pet to the vet unless being given direct, written permission to do so. If you do, you run the risk of paying the bill yourself as well as the owner being mad at you for taking her without permission.
Emergency situations are the only time when such an action could be justified (bleeding out, severe allergic reaction, etc).
Really try to get in contact with the owner, and do not act until you do.