Jeremy Crawford has this to say on the subject:
One of the best uses of Disarming Attack is to knock something out of a foe's hands and then pick it up.
The rules don't stop a fighter from using Disarming Attack and picking up the dropped item. A DM might say the item is out of reach.
This might seem strange at first glance. Let's go back to the rules:
Moving Around Other Creatures
You can move through a nonhostile creature’s space. In contrast, you can move through a hostile creature’s space only if the creature is at least two sizes larger or smaller than you. Remember that another creature’s space is difficult terrain for you.
Whether a creature is a friend or an enemy, you can’t willingly end your move in its space.
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.