There are certainly many important moral and pragmatic arguments about the propriety of citizens' use of deadly force to defend property or to restore order, especially when the normal go-through-the-proper-channels means of protecting property or deterring crime are absent. Texas law (see secs. 9.41-9.43), for instance gives citizens a fairly broad right to use deadly force to defend both their property and, in many instances, others' property, even under normal conditions.
Based on my quick glance at the statutes, though, I doubt that even the Texas defense would extend simply to shooting looters, chiefly because it requires an attempt to protect or recover (recover for the benefit of the owner, I assume) specific property. The shooting of looters of others' property isn't really aimed at or likely to recover that property for the owner's benefit. Even if the looter can't get away with the TV set, it's unlikely that the owner of the TV set will get it back that way; and shooting for the purpose of generally deterring others' misconduct (as opposed to, for instance, shooting for the purpose of specifically deterring people who are right near the target from looting that particular place) probably isn't covered under Texas law. Moreover, there is still a moral question as to the propriety of such shooting, and the practical matters that Orin raises (and I lean towards sharing Orin's concerns).
But in any event, that's Texas law, not Louisiana law; and it might be worth remembering that, under Louisiana law — which generally (with some exceptions not applicable here) doesn't allow the use of deadly force to defend property — the shooting of looters by private citizens is a crime: the crime of murder. Some might argue that it shouldn't be a crime, either because the Texas rule (or an even broader version of it) is generally right, or because when civil authority collapses citizens have to be able to protect their property (if they're shooting people who are looting their own property) or try to restore order more broadly. And that's all well and good for blogs. But if you are ever put in this position yourself (and I hope none of us ever will be), you might want to keep in mind that, in most states and probably even in Texas, the current criminal law is very much not on your side.
Comments: Please be calm and civil, and if you want to make arguments about the current state of the law, please point to specific legal authorities. In particular, if you want to argue about Texas law, please make sure you have carefully read the statutes I cited.