For individual appraisers and the company they work for (as appropriate), my credit union verifies the appraiser license is current and checks for adverse actions by the state licensing body, checks their BBB rating/complaints, checks state court records to see if they've been the defendant in a lawsuit, checks OFAC, get a W-9, and tracks their E&O coverage expiration. We mostly do the same for title companies, although they aren't licensed - we do require their privacy policy. We've had appraisers who refused to get E&O insurance, so we dropped them from our appraiser panel.
With the increase in fraud related to title companies being hacked/email takeover, etc., I'd love to require info about their security procedures, but I expect the majority of them would have nothing to give us.