I have seen a similar issue in a different vehicle that was caused by a bad motor mount and a really bad transmission mount.
See if you're getting a lot of movement in the engine/trans when someone shifts between drive and reverse over and over. If you're seeing nothing, watch from the side while they hit the gas hard from a stop in both directions. This is also a great way to test if you're getting axle wrap/wheel hop.

Have you been testing for play in drivetrain parts in park or in neutral? sometimes you won't be able to tell that parts are bad while it is bound up in gear. Obviously use tire chocks before you go to test this, but that is what I would do to test it all if you haven't tried it before.

If nothing is showing up, buy a cheap action camera and mount it pointing at the drivetrain parts or rear axle. Drive at several different speeds to replicate the issue. When all else fails, this can be a great way to see what might be going on.