What are your legal rights if the house you bid on a home and the appraisal falls short of your offer?

Does your contract contain a contingency for you to obtain financing? If so, then it is unlikely that the lender will give you a mortgage commitment if the house does not appraise for the full purchase price. The loan-to-value ratio they use is based on the appraised value rather than the purchase price. If you cannot get a loan then the contingency means you can cancel the contract and get your deposit back.