That would depend on the type of proof you have. Just try to work things out with your sister-in-law.
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Take copies of all your documentation to the vet. If you can prove the dog they treated is yours, you should be able to get the records.
However, how is it your sister - in - law took YOUR dog to the vet's?
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The above attorneys are correct. You can go to the veterinary hospital with as much proof as possible that it is your dog and request the records. The easier way will be to have your sister-in-law put your name on the hospital record. Or you could hire an attorney to make the inquiry. It depends on how much this issue is worth to you.