Professor Vansittart, born at his father’s house in Great Ormond Street, London, was the second son of Arthur Van Sittart of Shottesbrooke Park in Berkshire, by his wife Martha, eldest daughter of Sir John Stonhouse, bart., of Radley Hall, also in Berkshire, Controller of the Household to Queen Anne. Henry Vansittart, the Governor of Bengal, was his younger brother.

Robert was educated at Reading and at Winchester. He matriculated at Trinity College, Oxford, on 8th April 1745, was elected a fellow of All Souls' College and graduated BCL in 1751 and DCL in 1767. In 1753, he was called to the bar by the society of the Inner Temple. On 17th May 1760, he was nominated as Recorder of Monmouth, in 1763 Recorder of Maidenhead, in 1764 Recorder of Newbury, in 1765 of Maidenhead and, in 1770, Recorder of Windsor. In 1767, he was appointed regius professor of civil law in the university, a post which he held till his death. For some years previous to his appointment, he performed the duties of public orator for his predecessor, Robert Jenner.

Vansittart was on intimate terms with the painters, George Knapton and Hogarth, as well as with the poets, Paul Whitehead and Cowper. In Italy, he met Goethe, who named a character in one of his comedies after him. He was a friend of Dr. John­son, who regarded him with much affection and who was invited to visit India with him by his brother, Henry. In 1759, in a festive moment, Dr. Johnson, while on a visit to Oxford, proposed that they should scale the walls of All Souls' together. On another occasion, while Vansittart was edifying Boswell with a lengthy story of a flea, Johnson burst in with “It is a pity, sir, that you have not seen a lion; for a flea has taken you such a time that a lion must have served you for a twelve-month.”

Vansittart, who was elected a fellow of the Society of Antiquaries on 4th June 1767, amused his leisure with antiquarian studies. In the year of his election, he edited 'Certain Ancient Tracts concerning the Management of Landed Property', which consisted of reprints of Gentian Hervet's translation of 'Xenophon's Treatise of the Householde' (1534); Sir Anthony Fitzherbert's 'Boke of Husbandry' (1534); and Sir Anthony Fitzherbert's 'Surveyinge' (1539).

Vansittart was a man of licentious and debauched habits and, like his brother, Henry, was a member of the 'Franciscans of Medmenham’ otherwise known as the 'Hell Fire Club.' To this society, he presented with great pomp a baboon sent from India by Henry, to which Sir Francis Dashwood was accustomed to administer the eucharist at their meetings. Vansittart died at Oxford, unmarried, on 31st January 1789, and was buried in a vault in the Chapel of All Souls' College. In person, he was tall and very thin, and the members of the Oxford bar gave the name of ‘Counsellor Van' to a sharp-pointed rock on the River Wye from a fancied resemblance.

Two portraits of Vansittart exist. One by Hogarth represents him as a young man, with a kerchief in the colours of the 'Franciscans' wound in turban fashion over the head, embroidered with the motto 'Love and Friendship’. The other, painted by Sir Joshua Reynolds, depicts him in later life.