Their insurance is not your problem. Send them a demand letter immediately enclosing the vet bills. If they do not pay, sue them in small claims court. Your sister must testify in court, and you must take the vet bills with you. Whether or not the neighbor wants to tender it to his/her insurance is not your problem. That's between the neighbor and the carrier. If you win, the court will issue the judgment and your neighbor will have to pay you, where he gets the money will not be your concern legally. You can also have a lawyer send a demand letter to get his attention.
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As the prior attorney stated, you want to initially send a "demand letter" requesting that the people pay the bill with a deadline to respond. You may want to hire an attorney to do this. If they do not respond and/or you are unable to work something out, then you will either want to sue them in small claims court or hire an attorney to sue them. Yes, you can seek reimbursement. The dog owner's home owners insurance may cover the expenses.