So I just did a basic google search for the question "Who does a real estate appraiser work for?" and got this:
"The mortgage lender or financing organization typically hires the appraiser. "
Assuming that's true in all cases (obviously a Google search is not real "legal research"), although it's clear that YOU suffered a compensable injury by overpaying for the house, the question would be whether you have to file a claim against the lender or against the appraiser.
If you didn't hire the appraiser (either directly or through a POA arrangement) then the appraiser may have owed no legal duty to you, the buyer. His duty was to the lender.
Certainly the lender is going to be pissed that they loaned money on a house for more than the house was worth, but the question is going to be this: If you relied detrimentally on the appraisal done by an agent of your lender and paid too much, do you have recourse against the lender for the negligent act of their agent, the appraiser? Or does the law presume you to be sophisticated enough to be responsible for negotiating a price that is agreeable to you and to look out for your own interests?
From a legal standpoint the question would be whether you, the buyer, are entitled to rely on the representation of the appraiser. That may be controlled by contract (your contract with the lender), or by statute, or by case law in that state.