He was the son of John Vesey by his second wife Anne, daughter of Colonel Agmondisham Muschamp.
He married firstly, Charlotte, daughter of William Sarsfield and Mary Crofts, and an alleged grand-daughter of Charles II and Lucy Walter.
Vesey's second wife was Jane, daughter of Captain Edward Pottinger and widow of John Reynolds and of Sir Thomas Butler, 3rd Baronet.
[1] He was succeeded by his son Agmondisham Vesey who inherited Lucan Manor.
The name Agmondisham originally comes from an archaic spelling of the town of Amersham in Surrey, England.