Your best approach is to deal directly with the humans involved, first.

If you think the veterinary hospital was at fault, because it used faulty cages or had incompetent help, for example -- ask for what you feel is a fair amount to compensate you for your loss.

If reasoning fails, you could go the legal route -- a small claims court lawsuit -- but it's not likely to be worth it. You would have to prove that the hospital was unreasonably careless. You will need some good evidence to do this, such as testimony from a staffer establishing the lax behavior. And the law deals in monetary, not emotional, losses. You may not be able to persuade a judge that you suffered much of a loss, in financial terms. Some judges will take into account the sentimental value of a lost pet, but many won't.