It's a moot point whether The Misfit really seeks justice as such. But there seems little doubt that he has no hesitation in putting himself forward as judge, jury, and executioner, especially in relation to the grandmother. We gather from The Misfit that he's mightily aggrieved, bitter at the hand he's been dealt in life. By killing the grandmother and her family he seems to feel that wrongs are being righted somehow. In that sense, he does have a moral code of sorts, albeit one that all civilized people would find abhorrent. He also has a sense of justice, though again not one that most people would recognize or accept.
One way of looking at The Misfit is to say that he pursues what he considers true justice. But he can only do this by committing a conventional act of injustice. True justice in this case means original sin, something in which the Augustinian Catholic O'Connor strongly believed. The Misfit acts as an unwelcome reminder of the universal sin in which, according to O'Connor's brand of Catholicism, we all share by virtue of our fallen humanity.
The grandmother's idealization of the antebellum South represents a desire to retreat to an Edenic paradise. Yet due to the disobedience of Adam and Eve humankind was expelled from paradise, forced to wander the earth and carry the stain of original sin for all eternity. There is to be no going back. The Misfit reminds us of that in the most brutal way imaginable. He is an Old Testament dispenser of justice, sent to warn man of the dangers of pride and of the folly of harking back to a romanticized past before that first disobedience in the Garden of Eden.