My dog recently caught a virus and almost died were it not for our Vet. The Vet looked at our vaccination records and let us know that the people we took the dog to, to get vaccinated were not doing so correctly (they vaccinate every 2 weeks when they should be vaccinating every 3, as a result our dog caught the virus he should have been protected from), would it be possible to hold the business who vaccinated the dog responsible for the vet bills?
I'm not sure why someone other than your vet is vaccinating your dog, but you may have a negligence suit for the cost of your vet bills against these "people" who you took your dog to.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
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While I agree with my colleague, I think you are going to have to first look at the language of any contract you signed with the business who gave your dog the vaccines because I am betting that it limits its liability, and second, veterinary negligence actions are incredibly expensive to litigate and extremely time-consuming. You could try filing a claim with the business's insurer.