According to the AFI, there appear to be two major categories of Services.  One that is 'stand-alone' and one where the service is part of an acquisition program (such as Contractor Logistics Services for a newly-fielded weapons system.)

If the service is part of, or in support of, an acquisition program, those requirements are approved as part of the overall requirements for the program itself, so they would be part of/incorporated into  an Operational Requirements Document (ORD), Initial Requirements Document (ICD), Capability Development Document (CDD), or approved Acquisition Strategy.  The service is, in a sense, "embedded" in the requirements for the weapons system.  The Requirements Management community in the MAJCOM/A5/A8 that needs the capability develops those documents, then they are approved at the MAJCOM or HQ AF level, or Joint Staff level if the program is large enough.  Once there is a Material Development Decision, the acquisition community takes the CDD and translates it into a System Requirements Document that is put into the Request for Proposal to go out to industry for bid.  The services are then executed along with the materiel development, fielding, and sustainment.  This is all part of the JCIDS process for materiel.

If the service is stand alone, then the requirements development is the responsibility of the organization that needs the capability (and whoever is furnishing the funding to pay for it.)  IAW AFI 63-138; 3.5.4. The intent of the RAD process is for mission owners to validate the requirements for their organizations.  In the context of this question, it appears to be a Numbered Air Force (NAF), which has Wings underneath it that require the services.  There are not likely to be any acquisition people at the NAF level, so the task of developing the RAD would likely be taken on by the Requirements Management staff in the NAF  - probably in the A5 shop, or A8, if that exists at that particular NAF.  The RAD would then be approved by the NAF commander, or at the MAJCOM level if more than one NAF requires the services.   The Contracting organization at the appropriate level (NAF or Wing), then use the RAD to go out to industry to contract for the services.  The Contracting organization may also have a recommended format or template for a RAD that they can provide that should facilitate the contracting process.