There are a number of reasons why Candy doesn't shoot his old dog himself. He obviously has a powerful sentimental attachment. Slim doesn't ask him to kill his dog because Slim is apparently too kind-hearted to do so; but when Carlson volunteers to do the shooting, Slim is one of the men present who agree that the dog should be put out of its misery.
Candy could hardly shoot the dog because he doesn't own a gun. Besides that, he only has one hand. He probably couldn't handle a rifle at all. He would have to borrow Carlson's pistol, a German Luger, which may be the only handgun on the entire ranch. When the men all go riding off to look for Lennie after Curley's wife's death, they have shotguns and rifles but no handguns. But the Luger is an exotic foreign weapon, and Candy could have a terrible time handling it and the dog while attempting to kill the animal--especially since he would be reluctant to pull the trigger anyway.
The only practical way to shoot the dog is to hold it still with one hand and use the other hand to aim the pistol and pull the trigger. Carlson knows exactly what to do.
From his pocket Carlson took a little leather thong. He stooped over and tied it around the old dog's neck. All the men except Candy watched him "Come boy. Come on, boy," he said gently....He twitched the thong. "Come on, boy." The old dog got slowly and stiffly to his feet and followed the gently pulling leash.
We don't see the execution, but we understand that Carlson held the dog tightly by the thong so that he could take careful aim at the back of its head. Candy could never have handled such a job, in spite of the fact that he blames himself for not killing his own dog. He could have botched the killing and only wounded the dog or wounded himself in the leg. An old man with one hand has no business handling firearms.
When Carlson asks Slim to tell Candy he should shoot his old dog, it shows how much authority Slim has among his fellow workers. Carlson finally has to try to persuade Candy himself. He meets with considerable resistance from the old swamper. Finally Carlson has to appeal to Slim to second him. Slim's agreement with Carlson seals the dog's fate.
"Carl's right, Candy. That dog ain't no good to himself. I wisht somebody's shoot me if I got old an' a cripple."
Candy looked helplessly at him, for Slim's opinions were law.
In Chapter 3 of Of Mice and Men, Carlson enters the bunkhouse, complaining about losing to Crooks in the game of horseshoes. Then, he sniffs the air, sniffs again and looks at Candy's dog. "Godawmighty, that dog stinks. Get him outa her, Candy!" Candy, who loves his dog, replies defensively,
"Well-hell! I had him so long. Had him since he was a pup. I herded sheep with him....You wouldn't think it to look at him now, but he was the best damn sheep dog I ever seen."
Carlson is not to be put off. He argues that the dog suffers and should be put out of his misery and offers to shoot the dog himself. Candy sits up and scratches his white whiskers. "'I'm so used to him, he said softly, I had him from a pup.'" Candy simply cannot bring himself to shoot the old, feeble dog that has been his companion for years. Furthermore, he sees something of himself in this old dog. Someday he, too, may be thrown out, Candy worries.