Private Lenders rely primarily on the value of the property in making loan decisions. So it follows that they are more particular about whom they’ll accept appraisals from.
It is always a good idea to ask the interested lender for their approved appraisers for a given location before ordering the appraisal. If the borrower pays for an appraisal from an appraiser not on the approved list, it is likely that the cost of that appraisal will be wasted as a new appraisal from an approved appraiser will be required. If they’re lucky the lender may make an exception (for example if the LTV is low), but that’s not guaranteed.
The good news is that most private lenders have similar preferred appraisers. So if the 1st lender doesn’t like the appraisal, it may still have value to other lenders interested in the deal.