Jayne Hickey, 30, and her son Christian, now aged eight, were both shot in the leg when two men knocked on their front door in Salford.
Police said both are continuing to recover after numerous operations.
The shootings were linked to armed gang feuds in the city including the murder of Salford's "Mr Big" Paul Massey.
Two men knocked at the door of the Hickey's family home on Gillingham Road in Eccles on October 12 last year and after reportedly asking for the boy's father, they fired shots.
The men were then seen speeding away from nearby Dover Street in a black Audi S3 shortly afterwards.
Det Ch Insp Howard Millington said: "Christian and Jayne are recovering well but what happened to them will affect them for the rest of their lives and we are determined to bring them justice."
He said officers want to speak to a youngster on a motorbike the offenders passed in the Audi on Worsley Road.
The men "drove at high speed" close to the rider, "causing him to pull over".
Officers are also trying to trace the occupants of a small dark-coloured hatchback saloon which had a broad stripe running diagonally from the front door over the wheel arches, that the Audi also passed.
The first offender is around 19 or 20 years old, of a "chunky" build and with a Salford accent, while the second man is only described as white. Both were wearing baseball caps.
So far 18 arrests have been made over the feuds between organised crime gangs in Salford. However, no arrests have been made over the murder of Mr Massey, who was shot dead outside his home last July.
A renewed appeal to find the men who shot a seven-year-old boy and his mother on their doorstep a year ago has been launched.