While it is rather difficult to find parallels between The Story of Edgar Sawtelle and Hamlet in each chapter as the narrative is only loosely based upon Shakespeare's famous play, there are certainly comparisons that can be made between the two narratives. 

• Both lack voice

For Edgar his is a physical problem as he has been born with an inability to speak; for Hamlet, his protestations to his mother regarding her lustful nature that has led her to marry his uncle go unheeded; later, Polonius accuses Hamlet of being mad, so he has no recognized say in what has transpired in the state.

• Claude, like Claudius of Hamlet is deceptive

In Part I, "The Stray," Edgar is witness to Claude's surreptitious attempt to kill the stray German shepherd Forte. When Claude lies to his brother that he shot the deer whose leg the dog had mangled, Claude gives Edgar a threatening look, so the fearful Edgar has to lie to his parents that he was not witness to anything. 

• There is animosity between the brothers Claude and Gar, and eventually Claude kills Gar.

Claude, who has been released from prison argues frequently with his brother until one night they physically fight for the keys to the truck. Losing, Gar tosses the keys on the ground and Car drives off with the truck. Later, Gar takes Claude's bedroll and things to town where Claude has procured work.

• Claude kills his brother Gar, just as Claudius kills King Hamlet by poisoning. 

Edgar discovers his father lying on the cement floor of the barn, but his father dies before anyone can come because he cannot communicate with the telephone operator because he is mute. There are no witnesses, so all Edgar can do is suspect his uncle.

• Uncle Claude becomes Trudy's lover, just as Claudius marries Hamlet's mother

When the dogs engage in a brutal fight, Edgar's mother Trudy phones Claude in desperation. He comes to her aide, and ingratiates himself,-"one may smile, and smile, and be a villain" (Hamlet 1.5.108)-- becoming her lover.

Not long afterwards, Edgar awakens and hears the dogs barking. As he searches in the storm for the cause, Edgar sees the ghost of his father (like the ghost of King Hamlet), who communicates to him through signs until the boy discovers a syringe, just like the ones Gar has used on the ailing dogs.

• Just as Hamlet has a play staged re-enacting the poisoning in the ear which Claudius performed on King Hamlet in the garden, after confirming that his mother is, indeed, romantically linked with Claude, Edgar re-enacts what he believes has happened between his father and his uncle by teaching the dogs to role play in a similar act.

Claude blanches in guilt as he watches, and Edgar raises a pitchfork to kill him, but accidentally strikes Dr. Pagineau, the old vet, slaying him in a similar manner that Hamlet strikes through the curtain at Polonius. Edgar flees into the woods with three of the dogs. In Hamlet, the Prince goes to England.

• Thoughts of Edgar mirror those of Hamlet. For instance, at the beginning of Part II, Edgar thinks,

Chaos of image and memory. Her touch had released a song in the tiny increment of the poison bound up in him that would...ripen to sorrow.....
The sleep that followed was black, not at all contained in it.

This passage recalls the soliloquy of Hamlet in which he contemplates suicide, thinking it better 

To die, to sleep
To sleep, perchance to dream, ay there's the rub (3.1.62-63)

• Edgar returns home only to engage in battle as Hamlet does on his return from England.

On his way to a commune in Canada that he has seen advertised on television, but delayed when one of the dogs is injured, Edgar seeks help from a man named Henry. Later, Henry and Edgar are turned back on their way to Canada by a tornado, so Edgar decides to return home. As Hamlet returns from England only to find himself engaged in a duel with Laertes, a duel constructed by the evil Claudius, Edgar finds himself trapped in the barn by Glen, the son of the vet that Edgar accidentally killed, and Claude. Edgar tosses ether into the eyes of Glen to deflect his attack; however, the ether hits the oil lamp and ignites a fire. Edgar tries to rescues his father's files on the dogs, etc., and Claude deceptively helps him since he has hidden the poison in the files.

This act parallels the Act V duel of Hamlet with Laertes in which his rapier has poisoned. When the swords are switched Hamlet impales Laertes and his mother drinks poison. In the barn, Claude tries to kill Edgar with the poison, but overcome by smoke, he dies alongside Edgar, just as Hamlet kills the deceptive Claudius, forcing him to drink the poison.