Most mortgages are made by mortgage originators who resell them. If I remember how it worked before the crash, the agent, mortgage originator, and title agent would work together to make the deal happen. There were some kickbacks. The mortgage originator would hire the appraiser directly. Now the appraiser is assigned by a 3rd party, with one consequence being that appraisers generally don't know the nuances of local markets and are unable to properly price in a hot market.
Appraisals are made by the lending institution. That would be a very obvious conflict of interest otherwise.