Born the second son of Major General George Henry Vesey and educated at Wellington College and the Royal Military College Sandhurst, Vesey was commissioned as a second lieutenant in the Queen's Royal Regiment on 20 February 1897. He was promoted to lieutenant on 20 October 1898. He served in the Second Boer War of 1899–1902, where he was wounded in the Battle of Colenso on 15 December 1899, and later served in the Natal from March to June 1900, including action at Laing's Nek in June. While in South Africa, he was acting adjutant of the 2nd Battalion of his regiment from 15 May to 29 November 1900. After peace was declared in May 1902, Vesey left South Africa on board the SS "Bavarian" and arrived in the United Kingdom the following month.