What could cause a D7000's mirror to flap and "Err" to be displayed when trying to take a picture?

Since you are getting a error on the display, look up the error in the user manual. If you need more information, contact Nikon customer support.

Depending on the error, it might be something you can fix yourself (bad memory card, low battery voltage), or it might require professional repair service.

Again, start with the user's manual.

This is most likely a power issue. If the battery supplies inadequate voltage to the camera, the camera may fail to do all tasks necessary to take a photo - and opening the shutter is one of the more power-intensive tasks.

If the battery was not already fully charged, charge it and try again. If that fails, buy a new battery for it, charge it fully, and try it again. If you are lucky, a camera store nearby may have a battery that fits that you can use for testing. (If so, buy your battery there, if your test is successful, as thanks to the store for letting you experiment.)