Q:

Does the Wii/Wii U sensor bar need to be plugged into the console to function?

Is the Wii/Wii U sensor bar a receiver or simply a transmitter? I'm running into the problem where I can't just run the sensor bar cord across the entire room to the screen, and I don't want to buy a 3rd-party wireless sensor bar because my experience with 3rd party stuff has been terrible.
Is there an easy way to supply power to it, if it is just a transmitter?
My theory is that the Wiimote passes information to the Wii/Wii U, not the sensor bar, but I really have no idea.

A:

The Wii/Wii U sensor bar simply emits a set of infrared lights that the Wii Remote can detect. The cable that connects the sensor bar to your console is solely used to supply power; without power, however, the sensor bar will not function.
There do exist third-party wireless sensor bar products that use batteries, but since you say that you're not interested in third-party products, you may be out of luck on that front.
I'm not aware of the existence any sort of adapter product that will convert the sensor bar's plug connector into other connector formats, but I did happen to find a video where someone demonstrates doing so as a homemade project. However, it involves wire splicing and that sort of dirty work:

