My researches had reached the stage where there was a stong possibility that James Moon had travelled on a ship from Bristol England, and I was considering making a visit to the Bristol Records office to look at the early Quaker records.
However, as you already appear to have far more information than your original post suggested, could you please indicate whether you have already searched the Bristol, Gloucestershire, or Somerset Monthly Meeting Minutes, or the Bristol and Somerst Quarterly Meeting minutes for any record of James's departure, before I make the journey.
The Records Office also carries copies of some early Ship's Manifests, although they are often incomplete, particularly where a ship was chartered for transport rather than taking fare paying passengers. There isa further problem identifying many ships, as the Quakers sometimes renamed the ship to something that was acceptable to them for the journey, although this may have only been a later practice to disassociate themselves from ships that were engaged in the Bristol Slave trade.
I know from surviving contemporary correspondence, that one group from South West England re-named their ship "The Friend" for their journey from Bristol in the early 19th century. This renaming may have taken place at sea because there is no record of a ship of that name ever leaving the Port of Bristol.