Although temporarily assigned to several projects, it now appears that the designation F-109 was never actually used by any aircraft.
In 1955, the McDonnell Corporation proposed that the designation F-109 be assigned to the two-seat all-weather interceptor variant of the Voodoo. The Air Force turned down this proposal, and the aircraft was assigned the designation F-101B instead.
Throughout the 1950s there were published reports that the F-109 designation had been assigned to a vertical-takeoff aircraft designed by the Ryan Aeronautical Company. However, this aircraft was actually designated X-13 (a designation in the X-for-experimental series). The X-13 was strictly experimental and was never intended as an operational fighter aircraft, and it never actually bore the F-109 designation.
Many references that I have read state that the F-109 designation was assigned to the Bell D-188A, a late 1950s private venture proposal by the Bell Aircraft Corporation for a Mach 2+ V/STOL fighter. This proposal called for a high-winged aircraft powered by eight General Electric J85-GE-5 turbojets. Two of these engines were mounted horizontally in the rear fuselage and were fed by cheek-type air intakes mounted on the sides of the rear fuselage. Two other J85 engines were mounted vertically in the fuselage behind the pilot's cockpit. They provided lift during vertical takeoff and landing, but were shut down for ordinary horizontal flight. The other four engines were mounted in two pairs in movable pods at the wingtips. The pods were rotated into a vertical position for vertical takeoff and landing, then were rotated horizontally for level flight.
The project had gotten as far as the mockup stage when, in February 1958, the Bell Aircraft Corporation requested that the USAF assign the designation XF-109 to the D-188A project. The Air Force had no interest in the proposal and turned down the request. Consequently, the D-188A never, in fact, received a USAF designation, although the USAF serial numbers 59-2109 and 60-2715 have been associated with this project (although these serial numbers were actually assigned to an IM-99 Bomarc and an AGM-12 Bullpup missile respectively). The mockup of the D-188 was actually adorned by a fictitious serial number of 59-2109. In the event, the D-188A never did find favor with the military, although the general concept was later taken up by West Germany in the E.W.R-Sud VJ 101C.