The novel does not clearly indicate how Edward would have received the letter. For that reason, a number of possible explanations come to mind. Remember that Edward does have the ability to read minds. While he cannot read Bella's mind, he can access the thoughts of others.
He could read Alice's or Jasper's minds, but since they are under the impression that the letter in their possession is for Bella's mother, it is highly unlikely Edward would have learned anything profitable from peering into their thoughts. Since Edward does arrive in Phoenix, however, depending on the proximity of Bella's location to the airport in Phoenix, Edward could perhaps read James's mind. Being with Bella, James's thoughts would offer Edward a constructive picture of her location and he could go from there. Edward, meeting Alice and Jasper in the airport, could have volunteered to deliver Bella's letter to her mother for Alice. Though it is not likely he would open it, perhaps in his attempt to deliver the letter, he stumbles upon a clue as to Bella's whereabouts.
Since Twilight is written from Bella's perspective, a definitive answer to this question will not present itself. Since Midnight Sun is apparently Twilight written from Edward's perspective, perhaps it could also provide some clues to this question.