Do you know where James Moon lived before setting off for America.
The UK Quaker Composite Birth Marriage and Death Registers (prepared after 1837 from earlier records) still exist and can be searched. Records of early members were often entered retrospectively, and are often incomplete. However if there were other Moon family members in the Society, who remained in the UK, their names may well appear in subsequent entries.
Most of the Monthly Meeting Minutes also still exist, although 1682 is very early for Quaker records to have survived. Any removal of James to the US would have been recorded in the MM Minutes and if the records have survived, would provide the "proof" you are looking for.