Why is LGA's Instrument Approach Procedure VOR-H misaligned with the runway?
Not all approached are aligned with a specific runway; some like this VOR approach into KLGA is not and just uses circling minimums.  

It is anticipated that the aircrew will, after reaching the MAP at MDA, visually confirm that they have the intended runway of landing in sight, thence circle to land on it visually at or above the circling minimums published.  The landing runway would be given by ATC or requested by the flight crew thence approved by ATC.

Nonprecision approaches with circling minimums only are quite common.  They are created this way only if:


  The final approach course alignment with the 
  runway centerline exceeds 30°. 
  
  The descent gradient is greater than 400 ft/NM from 
  the FAF to the threshold crossing height (TCH). When 
  this maximum gradient is exceeded, the circling 
  only approach procedure may be designed to meet 
  the gradient criteria limits. This does not preclude a 
  straight-in landing if a normal descent and landing 
  can be made in accordance with the applicable CFRs. 
  
  A runway is not clearly defined on the airfield. 


See the FAA’s Instrument Procedures Handbook.