Does the Orc's 'Aggressive' racial trait allow you to move with a grappled target as long as you move toward it?

You cannot end the move closer if you are already as close as you can get.

The second sentence of Aggressive says:

You must end this move closer to the enemy than you started.

If you are grappling the creature, you are most likely already as close as you can get to the creature in terms of how the rules measure distance between positions. Since you are as close as you can get, you cannot end your move closer to the enemy.

There is a way for this to work, but it requires a trip to hell.

First, suppose you did have a reach of 10 feet, as an orc, so were able to grapple from 10 feet.

Here's how this could work:

• Grapple the target from 10 feet away.
• Use your bonus action to use Aggressive.
• Move toward the target, but use the grapple rules for moving a grappled creature: "When you move, you can drag or carry the grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you."
• Use your last 5 feet of movement to move 5 feet toward the creature without moving the creature, thus ending your move 5 feet away from the creature.

You start 10 feet away, end 5 feet away, and are always moving toward the creature. But how can you get a reach of 10 feet?

Welcome to Avernus.

The devils of Avernus have weaponized the ichor that remains after a demon is slain. In Baldur's Gate: Descent into Avernus, we see:

When a creature comes in direct contact with demon ichor in Avernus, it must make a DC 10 Constitution saving throw. Fiends, oozes, plants, and undead automatically succeed on the saving throw. On a failed save, the creature is magically warped by the ichor, as determined by rolling on the Flesh Warping table.
[...]
Demon ichor is sticky enough to adhere to most weapons. A creature in Avernus that takes damage from a weapon coated with demon ichor suffers a flesh-warping transformation if it fails its saving throw (as described above). After hitting its target, the weapon is no longer coated with demon ichor.

One of the results on the Flesh Warping Table is:

The target’s arms become tentacles with fingers on the ends, increasing its reach by 5 feet.

So if you were an orc, and you went to Avernus, and started touching the puddles left by slain demons, you could possibly extend your reach by 5 feet, and be able to move creatures using the Aggressive feature.