He was born at Cherrington, Gloucestershire, in November 1679, and baptised there on 18 December 1679, was the second son of Joseph Trapp (1638-1698), rector of Cherrington from 1662, and grandson of John Trapp.
Early in his academic career Trapp began to versify for Oxford collections; and he wrote poetical paraphrases and translations which are included in the Miscellanies of John Dryden and Elijah Fenton.