This is probably impossible, but I wonder, if it were, whether there would be support for it or not.
Regarding greed, Thomas Aquinas wrote:
... greed for any temporal good is the bane of charity, inasmuch as a man turns away from the Divine good through cleaving to a temporal good.
Regarding vainglory, he also wrote:
... it is praiseworthy that a man should "take care of his good name," and that he should "provide good things in the sight of God and men": but not that he should take an empty pleasure in human praise. [...] wherefore it is said of the magnanimous man (Ethic. iv), that he cares not to be praised.
Even more, in the same section he states:
[Pope] Gregory [the Great] ... reckons pride to be the queen of all the vices, and vainglory, which is the immediate offspring of pride, he reckons to be a capital vice: and not without reason. For pride, as we shall state farther on (II-II:152:1 and II-II:152:2), denotes inordinate desire of excellence. But whatever good one may desire, one desires a certain perfection and excellence therefrom: wherefore the end of every vice is directed to the end of pride, so that this vice seems to exercise a kind of causality over the other vices, and ought not to be reckoned among the special sources of vice, known as the capital vices. [...] And since many vices arise from the inordinate desire thereof, it follows that vainglory is a capital vice.
Now, the Reputation System in SE, through the use of badges, reputation, and rankings, can induce users to sin via e.g. greed and vainglory. Naturally, this ultimately depends on each person, but does not deny the fact that SE might act as what John Paul II called a structure of sin, which calls for its removal/reform.
Upon this reading of the situation, it might seem logical that a significant amount of users might be interested in eliminating such system for Christianity.SE. Is this the case? Is this actually possible? Would this ultimately lower the quality of the Q&A the site get?