The Homestar Runner universe has several. Pom Pom speaks in bubbling sound effects. The Cheat speaks in animalistic squeaks and growls. Strong Mad originally spoke by yelling the word "DUH!" but eventually graduated to yelling intelligible words and short phrases. And Sherlock, the "star" of the "crazy cartoon" Sweet Cuppin' Cakes (who is described as looking like "a cross between a cow and a helicopter" despite actually resembling neither) only speaks in "these weird sounds, and no words." This is lampshaded in episode 2 of Strong Bad's Cool Game for Attractive People, in which Pom Pom reprimands Strong Bad for repeating everything he says.
Both Strong Mad and The Cheat are sometimes capable of speaking intelligibly. The Cheat, in particular, does the voices for all the "Powered by The Cheat" cartoons according to the creators—the voices sound like half-assed impersonations of the main cast, which are all delivered by the less vocally talented half of the two creators in reality.
There's also Homsar, who speaks English, but nothing he says makes any sense whatsoever. In episode 2 of Strong Bad's Cool Game for Attractive People, Strong Bad finally discovers a way to understand Homsar and speak his "language." If he imitates Homsar's usual speech pattern after this point, Homsar will ask him "Why are you talking gibberish, Strong Bad? You sound like a two year old boy!"
If we're talking about Homsar, then let's not forget Senor Cardgage.
"Carrageenan, Monteljohn. Can you detect me to the nearest bus stamp?"
There was a competition between Homsar and Senor Cardgage for the title of Non Sequitur Champion. Cardgage lost, and his prize for losing was having to complete a coherent sentence, which took him three tries. "Grape Soda Banked" is apparently as close as he comes to actual coherence.
The eponymous character of Spoilsbury Toast Boy. Though most of the characters' voices are garbled, they're relatively understandable. Toast Boy, on the other hand, speaks only in odd tones that require subtitles to be understood.
All the characters in Happy Tree Friends. The one who comes closest to making sense in speech is Pop.
The only other exception is Disco Bear, who says his Catch-Phrase "Oh yeah" and occasionally "Oh no" in normal English. He doesn't say anything else besides grunting, however.
There are only about 3 characters in Weebl & Bob whom the viewer can actually understand, which is even lampshaded twice. The only way the eponymous characters can understand each other is by reading their speech bubbles.
In Red vs. Blue, after being shot multiple times in the throat, Maine can no longer speak (not that he did much anyway) in anything other than a growl or snarl. Somehow Wash is still able to understand him just fine in season 8.
Ultra Fast Pony has multiple examples. Scootaloo only communicates by scatting (complete with translated subtitles). Diamond Tiara has a high pitch and a weird impediment (sort of a real version of the "ermahgerd" meme) which makes her difficult to understand. Bon Bon talks in an irritating, high-pitched "Ya ya ya ya!" (though she began speaking more intelligently in later appearances). Braeburn does speak English, but he's a Motor Mouth with a thick Irish accent, so other characters in the show have difficulty understanding him.
Mr. Malice of Bonus Stage is a strange case, he only ever speaks in music (not lyrics, but actual music), or other assorted sound effects (including the Mega Man Classic boss intro theme, the intro to Ludwig van Beethoven's 5th symphony, or Matt Wilson beatboxing), and is subtitled whenever he speaks.