There are many symbols in The Story of Edgar Sawtelle. Perhaps the most obvious one is Edgar's inability to speak. While he cannot verbally communicate, he is still able to successfully communicate with, and therefore train, the dogs. The overarching symbolism of this in the story is that even though he seemingly cannot communicate with them, his lack of speech actually allows him to communicate with the dogs at a deeper level.
Edgar's lack of verbal communication is a symbol for an ability to see more than others do; it's a symbol of looking beyond the everyday and finding something extraordinary. For example, consider when Edgar is unable to get help from calling 911 after his father falls in the barn. This shows us clearly how Edgar's being mute can be unhelpful. However, the symbolism for his being mute in this situation shows up clearly when his father's specter, if you will, shows up that night in the rain. The specter is able to communicate to Edgar and let him know that even if Edgar had been able to call, it would not have changed anything.
Since Edgar cannot speak, he is more attuned to the power words have. He uses this when he meticulously chooses names for his dogs--each dog's name is a symbol in itself, albeit (seemingly) randomly chosen from the dictionary.