Did Early American writers create an American literary tradition? If so, how?

If I understand your question correctly, you want to know if there are traits of American literature which distinguish it, say, from British literature or literature from another English-speaking country.

The first contributions to colonial literature were mainly logs and personal diaries relating the pioneers' experiences in the New World. At the same time, a kind of "lore" relying on word of mouth transmission was circulating, developing storytelling style and content influenced by tales of the indigenous people as well as European tradition. This was not intentional; rather it was a "happening" of circumstance and culture, as folklore goes.

Over time, a few traits emerged. One was "the tale tale," a mock-heroic kind of yarn flagrantly exaggerating the merits and deeds of its would-be heroes. Roots of this came from Indian lore, primarily Iroquois. (There are echoes of this, too, in such ancient European tales as Beowulf.) These tales are not told to be believed but rather to simply entertain the listener through hyperbole. Examples of this would be the tales of Pecos Bill and Paul Bunyan, respectively a cowboy and a logger from the Southwestern and Western frontier.

Another characteristic which developed was in the storytelling itself. In a very "campfire" style, the narrator would begin a narration, confuse the story line or the characters, back up and "try" to tell the tale again, and then get completely befudddled and "lost" along the way. The humour is in no deftly delivered punch line or quip at the end but in the false despair the storyteller feigns as he tried to relate his story to his audience.

If at first this invention was incidental (or perhaps accidental!), later writers such as Mark Twain exploited this burlesque style to the fullest. He used this twist of slapstick humour both in his written work and in public conferences, which were immensely popular at the time. Other writers remained faithful to a more refined European literary tradition of understatement, suggestion and play on words.

Other trends infiltrated the American literary scene, particularly African and Asian ones, with increasing immigration (voluntary or not!); but I will leave place for another answerer concerning these and other ethnic influences, OK? ("Okey" - African derivation!).