Does Cindy Sheehan Have Moral Authority?

No. What she may have is moral standing, analogous to legal standing. Legal standing is, roughly, the right to initiate a law suit. Here are the conditions for legal standing from some random legal web site:

There are three requirements for Article III standing: (1) injury in fact, which means an invasion of a legally protected interest that is (a) concrete and particularized, and (b) actual or imminent, not conjectural or hypothetical; (2) a causal relationship between the injury and the challenged conduct, which means that the injury fairly can be traced to the challenged action of the defendant, and has not resulted from the independent action of some third party not before the court; and (3) a likelihood that the injury will be redressed by a favorable decision, which means that the prospect of obtaining relief from the injury as a result of a favorable ruling is not too speculative.

Now, don't to be too literal about the legal analogy. Nobody thinks mothers of dead soldiers have a legal case against the President. What is clear is that Sheehan's son died in a war that is the result of Bush's executive decisions. How does she do, roughly, on the moral analogues to the conditions for legal standing?
(1) Has Sheehan been injured? I don't think we can dispute that a mother has a moral interest in the life and well-being of her children. And we cannot dispute that the loss of a son is injurious. But the fact that she has suffered a moral injury does not establish that someone has morally injured her. Which brings us to
(2) The conduct Sheehan has challenged is Bush's call as Commander and Chief to invade and occupy Iraq. It seems clear that Bush's commands are a causal factor in the death of her son. But one may fairly argue that the action that ultimately put Casey Sheehan in harms way was his decision to enlist in the military. Casey Sheehan volunteered to occupy a role that involves the risk of death. The fact that Bush called a war, one the indirect effects being that Casey Sheehan died, injuring Cindy Sheehan, is not enough to establish her moral standing in the matter.
This pushes us back to an element of (1). Is Sheehan's interest in the welfare of her son “morally protected” in a case where her adult child volunteered to expose himself to the risk of death?
My answer is that a volunteer soldier forfeits his morally protected interest to not be exposed to the risk of death through war when he becomes involved in a just war. He suffers no moral injury or injustice if killed in voluntary service in a morally justified war or conflict.
So the issue of Sheehan's moral standing just is the question of whether the Iraq war is just or unjust. For Bush to recognize that the death of Sheehan's son grants her special moral standing is to concede that the war is unjust. Which is why it would be politically insane to recognize her in particular, rather than simply sympathizing with the pain, and recognizing the sacrifice of the entire class of people who have lost loved ones in the war.
Now, I do think that the Iraq war is unjust. And I think that soldiers, and by extension their families and friends, who die in unjust wars clearly are the victims of injustice. And political and military leaders who are responsible for putting on unjust wars are the perpetrators of this injustice and so bear some moral culpability for these deaths. Because the Iraq war is not just, Cindy Sheehan's moral interest in the life of her son, via his own moral interest in his own life, remained morally protected, and so I conclude she does have moral standing. She is fully within her moral rights to demand a justfication and/or an apology from President Bush. I do think it's important to point out that her injury gives her no special moral authority. It simply gives her justified grounds for demanding some kind of rectification.