What is the current state of regulation regarding VTOL aircraft in the LSA category?
The FAA term for VTOL [other than helicopters and lighter-than-air craft] is "Powered Lift". Also of note, gyroplanes by nature may be able to land vertical but cannot takeoff vertical so are not VTOL.

LSA excludes powered lift. Not much more to it, unless maybe it launches from a catapult and lands with a parachute.

title 14 CFR section 1.1 "General definitions":


  Light-sport aircraft means an aircraft, other than a helicopter or
  powered-lift that, since its original certification, has continued to
  meet the following:
  
  (1) A maximum takeoff weight of not more than—
  
  (i) 1,320 pounds (600 kilograms) for aircraft not intended for
  operation on water; or
  
  (ii) 1,430 pounds (650 kilograms) for an aircraft intended for
  operation on water.
  
  (2) A maximum airspeed in level flight with maximum continuous power
  (VH) of not more than 120 knots CAS under standard atmospheric
  conditions at sea level.
  
  (3) A maximum never-exceed speed (VNE) of not more than 120 knots CAS
  for a glider.
  
  (4) A maximum stalling speed or minimum steady flight speed without
  the use of lift-enhancing devices (VS1) of not more than 45 knots CAS
  at the aircraft's maximum certificated takeoff weight and most
  critical center of gravity.
  
  (5) A maximum seating capacity of no more than two persons, including
  the pilot.
  
  (6) A single, reciprocating engine, if powered.
  
  (7) A fixed or ground-adjustable propeller if a powered aircraft other
  than a powered glider.
  
  (8) A fixed or feathering propeller system if a powered glider.
  
  (9) A fixed-pitch, semi-rigid, teetering, two-blade rotor system, if a
  gyroplane.
  
  (10) A nonpressurized cabin, if equipped with a cabin.
  
  (11) Fixed landing gear, except for an aircraft intended for operation
  on water or a glider.
  
  (12) Fixed or retractable landing gear, or a hull, for an aircraft
  intended for operation on water.
  
  (13) Fixed or retractable landing gear for a glider.