No, accounts payable departments do not fit the definition of a third party settlement organization.
Accounts payable departments are internal processors of payments within an organization and do not act as a third party. A third party settlement organization (TPSO) is defined as the central organization that has the contractual obligation to make payments to participating payees (generally, a merchant or business) of third party network transactions. Examples of TPSOs include apps used to handle money transfers between buyers and sellers. Since accounts payable departments do not operate as an external entity facilitating payments between third parties, they do not meet the criteria for being classified as a TPSO.
Sources:
FS-2023-6
Rev. Proc. 2013-26
PLR 201535008
FS-2024-3