All the non-native English speaking countries represented in Polandball
Generally played straight in The Order of the Stick. Races that get an Intelligence penalty frequently talk this way, and don't even get capital letters in print. This later gets a lampshade hung on it when we get to see a tribe of orcs in their private time, eager to learn better speech, because "gok look forward to first-person pronouns," among other aspirations.
In Gai Gin, the American protagonist's Japanese starts at the level of "No eat tomato," and improves to fluency during her prolonged stay in Japan. Her tortured, roundabout sentence construction is typical of new speakers. It fades and is replaced by comfortable if simple language, but her grammar remains off in varying ways and is prone to breaking down when she does. The effect is fantastic when compared to the competition. Granted, the competition is dominated by retarded kobolds.
In Narbonic, there is an awards convention interrupted by a giant robotic foot that speaks like this.
"MASTER ALSO RUN OUT OF FUNDS TO PROVIDE FOOT WITH ARTICLES OR PAST TENSES."
In The Law of Purple, we have Thud, who hails from the "primitive" Claw Jungles of Caligula. Subverted in that Thud and his tribe are all quite intelligent (not to mention far less racist than their supposedly "civilized" counterparts).
Parodied with Grammer Gorilla, "the super-strong simian who likes to talk good!"
Dr. Ironologue: You fools! Ending a sentence with a preposition sends Grammer Gorilla into an uncontrollable rage! Grammer Gorilla: Me hating you!
Also on the occasions when Torg becomes Jungle Torgo.
Orrig from Daughter of the Lilies. It's easy to see that his native language does not include the sound "w", much like German.
Orrig: (on his very first appearance) Here. Vhy Brent under rock?
In Digger, Ed the hyena is always being speaking like this thanks to his long solitude, but by that same token is extraordinarily philosophical and eloquent, in a broken sort of way.
In Gunnerkrigg Court, back in the days, Anja and Brinnie didn't quite speak in a perfect English. And in a non-English example, the one time we get a translation of Annie speaking Polish to Gamma, there are grammar errors present suggesting she's not fluent in Polish.
Damara Medigo from Homestuck talks like this when she isn't speaking in sexually explicit google translate-level Japanese.
The Tough's employer in the "Mallcop command" arc of Schlock Mercenary spoke with this kind of grammar, due to being raised among Galstandard Peroxide speaking Schuul. Or rather, being a copy of a Schuul personality in a human body. The Oafans in Book 14 also speak with Peroxide, though their language skills improve considerably after their brain upgrade at the end of that arc.
In White Dark Life Devin and Nessie both talk in incomplete sentences. Justified in that Devin has no understanding of speech and Nessie has to avoid talking too much or any man (or simple-minded woman) near her will go crazy.