No, you cannot provide the appraisal or a copy of it to the other bank unless state law requires that appraisals be transferable. The appraisal came with the prohibition, and you could be sued if you violate the prohibition. Explain that to your former applicant. Be willing to accept their request to come back to your bank for the loan.
For the future, if your bank wants to be able to transfer appraisals, be sure to make that requirement part of your appraisal engagement letter. Don't be surprised at the number of appraisers who will not accept such a provision.
John S. Burnett
Fighting for Compliance since 1976
Bankers' Threads User #8