He was born in Scituate, Massachusetts, and studied with Rev. James McSparran, missionary to Narragansett from the "Society for the Propagation of the Gospel in Foreign Parts", and with Rev. Nathiel Eells, of Scituate. He entered Harvard University (then known as Harvard College) at age 15, graduating in 1722. He preached at Windham, Connecticut, in 1725 and was ordained to succeed the Rev. Samuel Whiting as minister there in 1726, marrying Rev. Whiting's daughter Mary in 1727, and remaining 14 years, with a ministry marked by a rather severe orthodoxy (he once traveled to Springfield to oppose the ordination of a minister accused of Arminian tendencies).
He was learned both in theology and in science, and constructed the first orrery in America.
His religious views led to conflict within the school: he objected to the teachings of English minister George Whitefield, an itinerant minister of the Great Awakening, and other itinerant teachers such as Gilbert Tennent. Rev. Joseph Noyes, pastor in New Haven, invited James Davenport to his congregation to preach: Davenport used the opportunity to brand him an "unconverted man" and a "hypocrite": the congregation was eventually physically split, resulting in the two Congregational Churches that still stand on the New Haven Green.
In 1741, two masters' candidates at Yale were denied their degrees for their "disorderly and reckless endeavors to propagate" the Great Awakening, and the College made it an offence for a student to imply that the Rector, Trustees, or Tutors were "carnal or unconverted men" or "hypocrites". It was not long before a student, David Brainerd, did so, saying that Tutor Whittelsey "had no more grace than a chair", and was expelled. Jonathan Edwards, Rev. Aaron Burr (father of the Vice-President), and Jonathan Dickinson unsuccessfully appealed for Brainerd's reinstatement.
Clap campaigned for laws to inhibit itinerant preachers and lay exhorters, and to stop the disintegration of churches by separation. Religious disputation continued to fragment to student body, who refused to submit to discipline, avoided religious instruction from the "Old Lights" (preachers established before the Great Awakening), and attended separatist meetings. In 1742, Clap closed the college, sending the students home. He was supported by the General Assembly, and many of the more ardent students transferred to other institutions when Yale reopened in 1743.
Clap instituted Yale's library catalog in 1743, and drafted a new charter of the school, granted by the General Assembly in 1745, incorporating the institution as "The President and Fellows of Yale College in New Haven". Clap was sworn in as Yale's first President on June 1, 1745. His formulation of a new code of laws for Yale in Latin became, in 1745, the first book printed in New Haven.
Whitefield returned to New England to preach, and Yale issued "The Declaration of the Rector and Tutors of Yale College against the Rev. Mr. George Whitefield, his Principles and Designs, in a Letter to him". In 1746, Clap expelled Samuel Cooke from the Yale Corporation for his role in setting up the separatist congregation in New Haven.
In May 1747, the General Assembly granted Yale the right to hold a lottery to raise funds: this income, together with the proceeds from the sale of a French boat captured by the colony's frigate, were used to build Connecticut Hall, the second major structure at Yale. It was completed in 1753.
Meanwhile there were conflicts within the Corporation. Benjamin Gale, son-in-law of Jared Eliot, a Corporation member, had published a pamphlet arguing for discontinuation of the colonial grant to the college, and no grant was given in 1755. Clap set out to raise an endowment for a professorship of divinity, and Naphtali Daggett was appointed the Livingstonian Professor of Divinity on March 4, 1756. Noyes offered to share his pulpit with the new professor, agreeing to subscribe to the Assembly's Catechism and the Savoy Confession of Faith, and the students returned to his First Church for worship.
Clap, however, quickly became disenchanted with Noyes' conversion to orthodoxy and obtained a decision that not only could Yale students worship separately, they could form their own congregation and administer Communion. The announcement of the Corporation's decision on June 30, 1757, was bitterly controversial, and, in the aftermath, discipline at the College collapsed. The General Assembly intervened, ultimately siding with Clap.
The student body was caught up in the rebellious spirit of the 1760s, resolving to drink no "foreign spiritous Liquors any more" and declaiming in chapel against the British Parliament, and petitioning the Corporation with their grievances, insisting on the removal of the disciplinarian Clap. The students stopped going to classes and prayers and generally abused the tutors, who resigned.
The Corporation ordered an early spring vacation, and few undergraduates returned. President Clap resigned at the Corporation meeting in July, continuing in service until commencement in September 1766. Professor Naphtali Daggett followed him as president pro tempore.
Clap died in New Haven, Connecticut.