There are various options that let you make your enemy drop a held object.
For example the "Disarm"-Action Option from the DMG or the "Disarming Attack"-Maneuver from the Battle Master subclass.
Would I be able to pick up said dropped object from the feet of the target as my "free" object interaction?
Best Answer
RAI, yes
Jeremy Crawford has this to say on the subject:
This might seem strange at first glance. Let's go back to the rules:
The rules forbid moving through or ending your move in a hostile creature's space. You could interpret this to mean you can't invade a hostile creature's space at all. However, making an unarmed strike, as well as the Grapple and Shove attack options, necessarily requires invading their space and doesn't count as moving. It seems Crawford is treating picking up an item in the enemy's space the same way.
As one additional data point that this is probably the intent, Disarming Attack would be almost completely pointless if this weren't allowed. The enemy could simply pick up their item as their one object interaction on their following turn. It would take at least two creatures working in concert to deny the enemy his item - one to disarm and one to move the enemy - and both must succeed within the same round. You could attempt this multiple times with the Disarm rule in the DMG, but that's an optional rule.
Granted, Disarming Attack can provide some protection against opportunity attacks, but Goading Attack, Trip Attack and Maneuvering Attack do that job just as well or better.