The role of fate in J.R.R. Tolkien's The Hobbit is a bit complicated. Fate is traditionally defined as "the development of events beyond a person's control, regarded as determined by a supernatural power." There are many instances in the story of these kinds of events, and Tolkien relies heavily on prophecies, and their subsequent actions, to move the complicated plot forward.
At the end of the journey, Bilbo rather diminished the significance of the prophecies when he says, "Then the prophecies of the old songs have turned out to be true, after a fashion!"
But Gandalf replies:
“Surely you don’t disbelieve the prophecies, because you had a hand in bringing them about yourself? You don’t really suppose, do you, that all your adventures and escapes were managed by mere luck, just for your sole benefit? You are a very fine person, Mr. Baggins, and I am very fond of you; but you are only quite a little fellow in a wide world after all!”
Clearly prophecies--the laying-out of predestined things to come--do play a role in this novel, and that is evidence of fate.
On the other hand, chance and free will are obviously part of the novel, as well. For example, Bilbo tries to buy some time to think about the answer to Gollum's riddle and, by chance, answers the riddle. Bilbo has an uneasy feeling and insists that the others close the door; luckily they do so just in time to keep Smaug from entering; and Elrond just happens to look at and can read the map on the precise night on which the moon letters were written.
In the end, much of what happens in The Hobbit is based on prophecy; however, the characters do have choices (and Bilbo obviously makes his share of bad ones) and certain things do seem to happen purely by chance or coincidence. Gandalf usually has the last word throughout the novel, and when he says the prophecies have been fulfilled by the actions of man, that is a pretty good argument for fate as the controlling premise in this novel.