I have a 2006 Forester X with 219k miles.  A few years ago, I ran a couple of wires from the FWD fuse to a switch on the dash, so I can turn AWD off and on more conveniently when needed.  I have noticed recently that when AWD is off (FWD mode) there is a noticeable vibration when accelerating at 18-20 mph.  When I first noticed this in FWD, there was no vibration when AWD was on.  Now I feel just a little vibration when AWD is on.  No vibration when cruising or decelerating at this speed, no vibration when in neutral and revving the engine.  No vibration at 50-70 mph, which is what I would associate with out-of-balance tires.  Looking at the engine when someone has their foot on the brake and revs the engine showed some engine movement, but not a whole lot.

Any guidance on what the cause might be?

My first thought is the rear drive-shaft (prop-shaft) bearings. But these usually cause vibrations when transmitting torque, for instance when accelerating in AWD. Same goes for the wheel bearings and half-shaft bearings.  So it is unlikely that you'd feel any vibrations in FWD, with no torque to the rear end.

So that leaves the rear-diff.  I wonder if the rubber mounting-bushings on the rear-diff are failing. With no torque on the rear-end (when in FWD), perhaps the rear-diff can move around a bit, and this may be felt as a 'vibration'.

I dunno, this is just my best guess.