[Jim] then give the gun to Paul and told him to try his luck. Paul hadn't never handled a gun and he was nervous. He was shakin' so hard that he couldn't control the gun. He let fire and Jim sunk back in the boat, dead.
Doc Stair, bein' the coroner, jumped in Frank Abbott's flivver and rushed out to Scott's farm. Paul and old John was down on the shore of the lake. Paul had rowed the boat to shore, but they'd left the body in it, waiting for Doc to come.
Doc examined the body and said they might as well fetch it back to town. They was no use leavin' it there or callin' a jury, as it was a plain case of accidental shootin'.
[Idiom] take advantage of: to put to good use; to profit selfishly by; exploit.
I'm not certain your question is worded quite correctly. "To take advantage of" is an American idiom either (1) to put something to good use, as in to take advantage of a favorable opportunity, or (2) to profit selfishly by something or exploit something, as in to take advantage of someone's generosity or take advantage of someone's innocence.
As the excerpt above shows, no one involved in Jim's death exploited (i.e., used selfishly), selfishly profited by or selfishly used Jim's death. An example of selfishly using someone's death might be to gain from an insurance policy or in romance.
Perhaps what you mean is more along the lines of, "Who had the advantage of Jim Kendall's death?" In this case, the answer would be Doc Stair, Julie, and Paul Dickson. The phrase "had the advantage of" is an American idiom meaning to be put in a superior position.
With Jim Kendall's death, Julie found herself in a superior position because her tormentor was removed from her life. Paul Dickson found himself in a superior position as Jim could no longer torment him for his mental disability caused by a fall. Doc Stair found himself in a superior position because he no longer had to find a way of retaliating against Jim's cruelty to Julie:
It's a cinch Doc went up in the air and swore he'd make Jim suffer.
If on the other hand you mean something more like who found an advantage in Jim Kendall's death, the answer would be Doc Stair. It was Doc Stair who was Paul Dickson's confidant. It was Doc Stair who confided to Paul his opinion about Jim's cruel prank against Julie telling Paul that "anybody that would do a thing like that ought not to be let live." It was Doc who was concerned when he found out Paul had gone out rifle hunting with Jim. Therefore Doc is the only who had reason to suspect Paul didn't act by accident.
Doc is the only one who might suspect Paul intentionally aimed and fired at Jim. And Doc is the only one who had authority to declare Jim's death anything but an accident, because Doc Stair is also the coroner. Thus Doc Stair found an advantage in Jim's death: he held his peace about his suspicions and allowed both Paul and Julie some peace and happiness by calling the end of Jim a good riddance--however it came about. You can see that this raises two difficult questions of ethics and morality regarding (1) Paul's probable action and (2) Doc Stair's subsequent actions as coroner.