Absolutely, "The Story of an Hour" would be wonderful for storytelling competition. The audience will either know the story, and be waiting and reliving all the way to the twisted ending, or they will never have heard of it and be delighted with the ironic twist it takes. While the moral is there, it is not didactic. It waits for the listener to discover it.

I don't know how "moral" the story is, or if this is the kind of story you are looking for, but "The Story of an Hour" by Kate Chopin is very short, and has a nice twist ending.  The moral of the story is not a cut and dry type moral -- the audience would have to think a bit to come up with the lesson to be learned from it, but I think they would come to the conclusion that Chopin intended -- the ironic twist on "be careful what you wish for." 

Kids love the short book "The Dot" because it has such a great message. You could turn that into a short story somehow. What about a Native American creation myth? There are tons of these to choose from. "The Monkey's Paw" is also a great story, but it might be too scary for fourth graders.

What about an episode from The Petite Prince by Antoine Saint-Exupery?  This is a short novel about a pilot who crashes in the desert.  Stranded there, he is one day approached by a little prince from a small planet that he calls Asteroid 325.  This prince fell in love with a rose, but when it lied to him, he was desolate and decided to leave.  As he passes by other asteroids, he encounters many narrow-minded adults.  He meets a king, a vain man, a businessman, a geographer, and others; from each encounter, the small prince learns sound lessons about life.  These episodes can be used separately, if need be.

What about the Indian folktale of the blind brothers and the elephant?  (They all touch a different spot and try to guess what kind of an animal it might be and of course all have completely different ideas and get into a huge conflict about it.)  The moral is something about not allowing differences of opinion create conflict but rather coming together to arrive at truth in a harmonious way.

Good advice.  How about any of Aesop's Fables?  Most are quite short, but I don't know your time constraints so they might work.  I'm also always intrigued by stories with unexpected or surprise endings. Not all of them would be classified as moral stories, but they might work.  I'm thinking of "The Interlopers" by Saki (H.H. Munro), for example.  Anything by Saki or O. Henry might work for you if you're looking for surprise endings--again depending on your time limits.  Have fun! 

There are many from which to choose.  I think that the narrowing of options might be the most challenging element for you.  In making your selection, I would make sure that you select one that feels "right" for you.  Sometimes, the best way to proceed with storytelling assignments is more based on personal comfort with the material.  The way a story is told is directly contingent on the basic idea that one is familiar enough to reinterpret it with their own flair or "spin."  With this in mind, I suggest that finding a story with which you have comfort.  I have always thought the Greek Myths serve as great story telling elements.  They are able to bring the audience in quite nicely.  One such myth that is really good is the story of Athena and Arachne.  It holds some supernatural elements, but also has a great moral lesson of humility and understanding the need to possess respect for the powers of the divine.