Is it normal for visibility to be missing from the METAR?
My questions are:
  
  Is it really normal to see reports without viz?


Yes!  Well, maybe not "normal", but it does happen.  Automated weather stations are like another piece of equipment and components fail from time to time, including the components which measure the visibility.  The other components also fail from time to time (I've see ASOS/AWOS reports without temperature, barometric pressure, etc.)


  Why does this happen so often for AWOS when ATIS doesn't have problems
  with it?


You shouldn't see this at all with ATIS, because a human is involved, and if the equipment which measures visibility is inoperative, a trained observer (often the same person that is manually recording the ATIS) can look out the window and record the visibility themselves.


  Should this be reported to some entity (NOAA/FAA) to get these systems
  repaired or exchanged?


I would let the airport manager know just in case they weren't aware.  They are responsible for notifying the FAA.


  The fact that it is missing visibility information does not mean that
  visibility is unobstructed, right? I have heard people assume this. 


If the visibility is missing, it means just that.  It is missing.  It might be 1/4 SM or it might be 20 SM. Some people may make assumptions that if there is no precipitation, that the temperature / dew point spread is high, etc. that the visibility is good, but that is a potentially dangerous assumption because visibility can be influenced by many different things.  There could even be a dust storm coming through when everything else looks good!


  If the recording says "VISIBILITY MISSING", is it indicated in the metar
  somewhere?


Sort of.  It's just "missing" (i.e. the entire wind block from the METAR will not be there), so if you don't see it then, well, it's missing.