No — appointed representatives (ARs) do not need their own complaints procedure. The principal firm is on the hook for any complaint about regulated activities the AR carries out. In practice the AR simply uses the principal's procedure, and the principal has to oversee that they're following it properly. This matters because customers need a single, clear route to complain, and the FCA holds the principal accountable for fair treatment end-to-end.
Under DISP 1.1.5R and DISP 1.1.7R, a principal is responsible for complaints concerning the conduct of its appointed representative in carrying on regulated activities for which the principal has accepted responsibility (see s.39 FSMA 2000).
Operational consequences:
For internal guidance, refer to Anchor — Complaint Handling Rules and Anchor — Appointed Representatives: Complaints. Anchor is a living manual; check the current entry for the latest version.