I'm curious, because they obviously don't have those huge rotors pushing the air away, but they are still getting their upward thrust from somewhere.
My gut says, yes, but it is more difficult. Is this right?
The main problem with VRT is the airflow through the rotor which is disturbed because it is ingesting its own downwash.
The large area of air accelerated downwards keeps being sucked back in to the rotor because the helicopter sinks back into it. In this state the rotor provides very little thrust - it acts as a sort of divide-by-zero state, as the transition between normal flight and windmilling when the air streams upwards.
The flow field of the Harrier is different:
The defining problem of VRS is the lack of thrust from the rotor at downward velocities, this would not occur with the shrouded jet engine of the Harrier.
No. Considering that lift is created by the downward wash of jet exhaust as opposed to air flow through a powered lift disc, this phenomena isn't encountered here.
Harriers do have a problem called blowback where hot section exhaust can be ingested into the engine, causing a flameout in cases of high descent rates or where the landing pad becomes flooded with exhaust gases, causing them to be reingested into the engine.