Why can't we use orbital correlation diagrams for sigmatropic rearrangements?
The reason that comes up again and again is the one below, taken from Orbital Interaction Theory of Organic Chemistry. 


  Orbital correlation diagrams are useful for cycloadditions and electrocyclic reactions but not for sigmatropic rearrangements since no element of symmetry is preserved.
  
  Source:   Orbital Interaction Theory of Organic Chemistry. Arvi Rauk. 


In order to (easily) draw a correlation diagram you need to find some symmetry element that is preserved throughout the reaction. A correlation diagram is shown below for an electrocyclic ring closing reaction, note how the C2 axis is preserved during the process, allowing the various correlations to be made. 


  
  
  Source: Dr Wallace's web notes on pericyclic reactions 


Since you mention Fleming's book, I'll quote him on this matter:  


  Step 3. Identify any symmetry elements maintained throughout the course of the reaction.
  
  Source: Molecular Orbitals and Organic Chemical Reactions: Reference ed. Ian Fleming. 


Essentially, you have to be able to look at the orbitals directly involved in the reaction (i.e. we dont care about random groups hanging off the side that play no role). 

In a sigmatropic rearrangement there is rarely (possibly never but I'm sure there are exceptions to the rule) a preserved symmetry element, which makes it impossible to draw a correlation diagram. 

There are, of course other approaches, each of which have advantages and limitations depending on the reaction in question:


Just apply the Woodward–Hoffmann rules
Use frontier orbital theory to look at the HOMO–LUMO interactions
Use a Zimmerman approach and look if the transition state is aromatic, or anti-aromatic.