Wishaw, Lanarkshire born wing half William “Tally” Sneddon began his football career with Wishaw Juniors and played for Rutherglen Glencairn in 1935, from where he joined Scottish League Falkirk in 1936. At Falkirk he scored 8 goals in 36 League appearances and appeared in a Scotland trial match, and he soon came to the notice of First Division Brentford, who signed him for £4,000 in June 1937, Sneddon making his Football League debut at Bolton Wanderers in August 1937. During two seasons at Griffin Park he scored twice in 70 appearances before a move to Swansea Town in July 1939 in a then-club record £2,000 deal along with Brentford teammate Sam Briddon.
The outbreak of the Second World War two months later put Sneddon’s professional career on hold after 3 matches at the start of the 1939-40 season, though he would continue to represent the Swans during the War. Sneddon played as a guest for Scottish clubs Ayr United and Hamilton Academical (4 appearances) as a guest during the Second World War. His most memorable spell came with a return to Brentford, helping the club to win the 1941-42 London War Cup. He returned to Griffin Park again during the 1942-43 season.
After the cessation of hostilities in 1945, Sneddon made three further competitive appearances for Swansea Town before leaving in November 1946 to join Newport County, where he made 18 appearances during the 1946-47 season before joining non league Welsh League side Milford United as player-manager. Sneddon later had a spell as manager of Burton Albion.