They also want the firearm used in the shooting to be recovered.
The teenager was found injured in Booth Meadow Court in the Thorplands area just after 7.30pm on Monday 30 September.
He suffered serious but not life-threatening injuries.
CCTV footage released by Northamptonshire Police shows two boys seen in the vicinity of the incident.
Lydia Patsalides, East Midlands Regional Manager for Crimestoppers charity, said: “Our charity believes everyone has the right to be safe."
A young person suffered serious injury and considerable emotional trauma after being shot and we would like to work with the community to find answers, to remove this firearm from the streets and the risk of whoever did this crime causing more harm to others.