Can an FHA-approved Mortgagee use non-employees as Loan Officers?

No.  The Mortgagee may not use non-employees as loan officers. 
The Mortgagee may establish a Sponsor/Sponsored Third-Party Originator (TPO) Relationship in which a Mortgagee (acting as the “sponsor”) permits another entity to act as an originator and originate Mortgages on behalf of the Mortgagee.
For additional information see Handbook 4000.1 Section I.A.5.a.v and I.A.6.j at

All policy information contained in this knowledge base article is based upon the referenced HUD policy document. Any lending or insuring decisions should adhere to the specific information contained in that underlying policy document.

Topic Number: KA-05121