In Lardner's "Haircut," who does take advatage of Jim Kendall's death?

[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.