And I'll have to give you a similar answer, MTI!
Our policy allows it, but make sure yours does. Gotta get it from the other bank or the appraiser. The customer can't walk in with it. See how it was ordered and make sure it fits with what your bank needs. If the engagement letter isn't in the appraisal report, ask the other bank. Since the appraisal is still in process, now would be the time to make sure the appraisal report will be what you need. Trust me....I've gotten appraisals from other banks and just had to shake my head. You don't want to wait a month to find out the other bank didn't order the value that you need!
Contact the appraiser and have the intended user section updated so that your bank can use the report. That may require a letter of assignment from the first bank. When we call to get the engagement letter, we usually get an assignment letter, too. Saves a second call after you've talked to the appraiser.
When you get the report, send it through your regular channels.
This feels like too much info....hope it helps without overwhelming.
And welcome to BOL!