Bombs kill by sound, if you want to look at it that way
Bombs kill via a shock wave. A compression of air due to the expansion of whatever that defines the bomb. If strong enough, things can be carried by that shock wave — like shrapnel. But, if the shock wave itself is strong enough, it kills and destroys. A nuclear bomb is a great example. After the radiation (in the first gazzillionth of a second), what happens next is a shock wave that flattens everything for some distance from the detonation.
And another word for a shock wave is sound.
What we hear with our ears is nothing more than a series of very gentle shock waves. Some, a single sharp wave (like the clap of hands); others, a continuous susurration of sinusoids (speech, or a tuning fork).
So, can you kill with sound? Yup, happens with bombs all the time.
but, what's doing the killing and what is being killed?
GrandmasterB and Burki correctly points out that Synalpheus pinkfloydi (known as the "Pink Floyd Pistol Shrimp," whomever named the little bounder must have just returned from a concert) can kill by snapping its claw near its prey. Very small prey. We humans can kill things that small, too, by clapping near them. Gnats, for example, can be killed via clapping concussion.
If, on the other hand, you want to kill a human (by means other than biblical elephant stomping and lion eating, different kinds of sounds were involved in that incident, like tummy growling and angry elephant honking... I love that story) then we need to create considerably more sound.
Fast and Furious
Part of the problem is that you need both amplitude and frequency. And you need a lot of both. A sharp explosion can be thought of as half a sine wave (it's more than that, but I'm simplifying). The amplitude is magnificent, but so is the frequency, demonstrated by how fast the sound level climbs from zero to its maximum amplitude. The lower the frequency (a slower climb to that maximum), the easier it is for a body to absorb it and the greater the amplitude must be to kill.
This is why bombs can kill but you little brother's stereo can't.
Possible but Impractical
Can sound kill you? The short answer is “yes” — and, rather shockingly, the European Space Agency says that it now has such a sonic weapon in its arsenal that, if it was so inclined, could kill you. (Source)
What the ESA has is a huge air horn used to test satellites against the noise (read: vibration) caused during launch. Stand in front of it and you die. But if you visit that site you'll see the horn is as tall as a human. And that's the problem. There's a lot of human to kill and we can often withstand a baseball bat to the head — and it takes a honking (pun intended) lot of sound to exceed the impact force of a baseball bat.
As the size of your target animal decreases, the likelihood that the source animal can create noise of high enough amplitude and frequency that it will kill increases.
As the size of your target animal increases, the likelihood that the source animal can create noise of high enough amplitude and frequency that it will kill decreases.
My guess is that the sound energy required increases exponentially with the size of the creature you intend to kill. Meaning you very quickly get to the point that the source animal can't create enough sound to kill the target. I suppose you could suggest that the foot of the elephant that killed that poacher constituted a shock wave of a sort — but consider the size and weight of the elephant compared to its prey! On the flip side, Synalpheus pinkfloydi might give us a nasty pinch, but the sound it uses to kill its prey simply sounds cute to a human.
Therefore, I'm going to say no. It's implausible that an animal could cause enough sound to kill a target animal. Unless you artistically consider the impact of an elephant's foot to be "sound." But if you want to use this as a super power, I might point you to The Spleen.