When I try to roll the rear tailgate window up or down from the key switch in the back the window doesn't budge and I can hear a buzzing sound from what I think is the relay box on the drivers side. When I try it from the switch in the front seat nothing happens and no buzzing sound.

I checked the rear wiper and its in its proper place, although I think it may have quit working the same time the window did.

I removed the motor from the tailgate and then turned the key switch in the back. The gear on the motor didn't spin and I heard the same buzzing noise, but no noise from the switch up front again.

SOooo, here is my question, do I have a bad motor or will the motor not operate if one of the sensors isn't contacting the right way.

The relay box arbitrates what can happen when.
You wouldn't want the washer and wiper to turn on when the window is down right?
Nor would you want the defogger to energize when the window is down.
Same with the motor it will not roll up if either of the side latches are open.

The buzzing is it a pulsating beep? That is usually what happens when something is tried when it shouldn't. Like rolling the window up with the tailgate open.

HOWEVER if you take a screwdriver and "lock" both latches then the window will roll out but if it does it MUST be supported as it will break the mounts in the rail.

SO with the above scenario you should be able to unplug the motor and see if the "buzzing" stops if it does then the motor may be bad.

I unplugged the motor and turned the key and it didn't make any noise. Although I didn't lock the latches because my window stopped working in the up position with the tailgate closed. Is there anyway to lock the latches with the window up and the tailgate closed or does this sound like a bad motor?

I was just looking over the diagnostics, does your window have a wiper and defogger?

If the tailgate is closed the latch switches are too.
Need to check the connections at the motor.
Just apply power and it should turn reverse leads and it should turn the opposite direction.

It does have a wiper and defogger.
The tailgate is closed so I'm assuming the latch switches are too.
The motor connections seem to be proper.

How do I apply power to turn the oppossite direction?

Just switch the leads, that is exactly what the relays do automaticly.
I sent you a message, I can send you more info that will help you track the problem down.