I'm not sure what to do.
I feel like I should at least be paid the $30 that it cost me to drive to the emergency vet but I'm not sure. The payment has already gone through. I haven't said anything to them yet.
What do other sitters do?
IMO, transportation for emergency vet visits come with caring for the dog. I'm assuming you're boarding or had the client for daycare? However, you can write off the mileage on your taxes.
I believe that a sitter should be paid for extra time, travel, or care. I call it a shuttle fee and it's included in my policies which every client is asked to read prior to service. If you want to add a statement about it for future clients, I would let them know upfront, yet probably let it go for this client.
I also agree, If the pet was in your care i would not charge an extra $30 to the client for driving. I personally feel that there is a huge difference from charging a client a pick up/ drop off fee (which is for the convenience for the owner) and having to take the pet to the emergency room.
I think in the future if that is something that you really do want to charge the owners, you should mention it during the meet and greet. " in the event their pet has to go to the emergency room you charge $30 for you driving their pet." (Although, I don't recommend charging the client, and I think you will have a lot of clients who would not be so happy with that charge. At least they will have the heads up vs spring it on them after the fact. )