There have got to be many more facts to this, for example why did you send the dog, did you actually give them the dog? What kind of a veterinarian would put down a dog brought in by someone not the owner. So, consult with an attorney to review the facts. Without all of the facts I can't offer an opinion. If someone did that to my dog there would certainly be a remedy that would be pursued correctly and properly through the legal system even if it were only for the "value" of the dog.
Law Offices of Andrew D. Myers, North Andover, MA & Derry, NH provide answers for informational purposes only. Actual legal advice can only be given by an attorney licensed in your jurisdiction, thoroughly familiar with the area of the law in which your concern lies. This creates no attorney-client relationship.
You can bring an action in small claims court without an attorney. Your main issues will be ascertaining the value of the dog (include your investment in vet treatments) and the lack of authority and basis for destroying your pet.
The foregoing is for general information purposes and does not establish an attorney-client relationship.
It is important to consult a dog/animal attorney in your area to determine what the typical outcomes are of cases in your area. Yes, you can take her to court. Small claims is likely to be your best option. You may want to look at filing a police report also.