Edgar was educated at Edinburgh Collegiate School, and at University College Hospital, London, qualifying M.R.C.S., L.R.C.P. in 1898. He took up a commission within the Medical Services, at the rank of Lieutenant, on 27th July 1899, then began active service in South Africa as Regimental Medical Officer with the 7th, 14th and 66th Batteries R.F.A., being present at Colenso on 15th December 1899. Shortly after he transferred to become Regimental Medical Officer to the Royal Inniskilling Fusiliers, serving at Vaalkrantz, Spion Kop, Pieter's Hill, and at the relief of Ladysmith. In April 1901 he rejoined the Artillery with the 28th, 78th and 66th Batteries and saw action at the battle of Roidam on 5th May 1900. In April 1901 he returned to the 1st Bn Inniskilling Fusiliers and remained with them until 1902. During this period Edgar was promoted to the rank of Captain, on 29th November 1900 whilst serving out in the field; was Mentioned in Despatches [L.G. 8//1901, 16/4/1901]; received the Queen's South African Medal with five clasps and the King's Medal with two clasps; and was awarded the Victoria Cross “On the 24th February 1900, Lieutenant Inkson carried 2nd Lieutenant Devenish (who was severely wounded and unable to walk) for three or four hundred yards, under a very heavy fire, to a place of safety. The ground over which Lieutenant Inkson had to move was much exposed, there being no cover available". [L.G. 15/1/1901].
Edgar returned to South Africa from 1902 to 1904. On 28th January 1910 he was promoted to the rank of Major.
On 28th October 1914 he was stationed at the War Office, having taken over as Deputy Assistant Director-General A.M.S. with the branch dealing with questions of expansion of hospital accommodation by private effort. He vacated this appointment in January 1915 and proceeded to France, entering the war on 15th January 1915. He became the Commanding Officer of the 2nd Field Ambulance, joining the unit at Bethune, and remained with the unit until 9th October 1916. During this time he had received promotion to the rank of Lieutenant Colonel on 1st March 1915.
According to the book ‘Commissioned Officers in the Medical Services of the British Army 1660-1960’ Edgar served with No 1 General Hospital in 1917, No 4 General Hospital from 1917 to 1919 and No 35 General Hospital in 1919, however, his name also appears in the war diary of No 46 Stationary Hospital as serving from 17th July to 13th August 1917, as well as in the war diary of the Matron-in-Chief as serving as Commanding Officer with No 4 Stationary Hospital around the dates of 13th December 1918 when she had visited the hospital, and 25th May 1919 when she reported “The OC, Lt. Colonel Inkson VC, was a patient, having ruptured a muscle in his leg playing badminton.”
In 1920 Edgar held the appointment of Assistant Director of Medical Services at Headquarters, Shorditch District. From there he became the Commanding Officer of No 82 General Hospital at Constantinople from 1920 to 1922. On 10th January 1922 he served under the rank of Temporary Colonel and held the appointment of Assistant Director of Medical Services at the headquarters, Colchester, until 1923, during which time he gained the rank of Colonel on 15th September 1922. From 1923 to 1926 he held the appointment of Deputy Director of Medical Services at the Fortress Headquarters, Gibraltar. He then retired on 15th September 1926, ceasing to be on the Roll of Officers on 5th April 1927. After he retired he lived at Summersdale, Chichester. He died in St Richard's Hospital, Chichester, He was the son of Surgeon General James Inkson, Army Medical Services