I'm having the same problem with my '06 V8 4Runner. Checked the fuse that protects the rear wiper and washer but it was good. Before I tear into the wiring did you ever find out what the problem was? Or if anyone can suggest where to look I'd appreciate it. Tnx.
Edit: jhumphrey, I just discovered the problem thanks to poster jacper in another thread. Apparently there is a sensor/interlock switch associated with the rear hatch window. So that when the window is down, the wiper and washer are disabled. By lowering and raising the window the problem was "fixed". Possibly to occur again in the future. I guess the sensor/interlock switch was somehow temporarily disabled, due perhaps to corrosion. Give it a try.
Edit 2: Here is the verbiage from the shop manual related to the rear hatch window-wiper/washer interaction: "Rear wiper operation: The back door ECU initiates rear wiper intermittent or continuous operation according to rear wiper SW command from the body ECU. At this point, the position of the back door window is confirmed and, if it is not fully closed, the command will be neglected preventing the rear wiper from starting. Rear washer operation: Turning the rear washer SW ON, with the ignition SW ON and the back door window fully closed, initiates rear washer operation."
Can you imagine that, a "back door ECU". Yikes. I grew up when ignition systems consisted of points, condenser and a rotor. Life was simple back then.