How much time has lapsed between letter 2 and 3.?

There are 112 days between March 17th and July 7th, which are the dates Walton includes in his correspondence.

We do not have an exact number of days or months that have elapsed. Letter 2 is written when he is gathering a ship and a crew to help him on his quest. The third letter, which is very brief, says he is now sailing and pursuing the goal that he has set for himself.