I had the same problem it's definitely a software issue because the camera would sometimes work in camera apps but now I could not switch cameras at all.
I may have found a solution of course I probably won't get credit for thanks to Apple.
I believe the problem arises from a bug when the camera is used in lock mode, try restarting your phone with the home and power button. Then open your camera in locked mode and set the flash to auto. Maybe take a few pictures, switch the camera back and fourth, turn HDR on and off, but make sure the flash is set to auto before you unlock the phone. Heck even try playing with the flashlight.
After I did this in locked mode to my surprise my camera and flashlight are now fully functional after I have unlocked it with the password. I'm not to sure what triggered this bug but when I first got iOS 7 and played with my phone some how I unlocked the lock screen without putting in a password. Just by using the camera and the notification and control Center, I unlocked the phone which might of triggered the bug.