My mail lady hit my dog who pays for the vet bill?

The mail carrier is considered a federal employee and would be covered under the Federal Tort Claims Act. You should file a claim with the local post office. You will need to provide evidence that the mail carrier was negligent and that your dog was injured as a result of the negligence. The post office may reimburse you for the cost of the veterinary bills, but you may need to sue the post office if they do not.