Yes, you can certainly use a YN-560 III off-camera from the Mamiya hotshoe in the way you envision with a Yongnuo transmitter. The single pin of the hotshoe simply means you can't use any TTL options and must go manual only.
The YN-560 III, btw, has two successor models: the YN-560 IV and announced (but not yet released) YN-660. They both offer a built-in RF master transmitter, as well as a receiver. The Mk IV can control three groups. The YN-660 will control six groups, and has an updated physical UI (dial instead of four-way buttons). For the stated purpose, here, however, there are no additional advantages to these later models.
The Yongnuo RF triggers that the YN-560 III's built-in receiver can recognize are the RF-602's transmitter unit [RF-600TX], RF-603, RF-603II, RF-605, and the dedicated YN-560-TX transmitter. Of these, the YN-560-TX is your best choice, since it's the only one that will let you control the power output level and zoom setting of a YN-560III/IV or YN-660 from the camera.