A gunman opened fire on a 33-year-old man in Dornoch Place, Bishopbriggs, at about 15:10 on Thursday.
The victim, who was collecting a child from St Helen's Primary, suffered non-life threatening injuries.
Detectives are treating the incident as attempted murder. Pupils were being offered support from social workers.
Officers said the gunman approached the victim, shot him and then ran off towards Ronaldsay Drive.
The suspect was holding a silver-coloured gun and wearing a high-visibility reflective yellow jacket with the hood pulled up.
He was described as being in his mid-40s, between 5ft 10in and 6ft tall and of heavy build.
Detectives said they were following a number of lines of inquiry.
They were keen to hear from anyone with information about a burnt-out grey Volkswagen Golf - registration FR12 PYB -  found nearby in Wood Lane which is believed to have been involved in the incident. It was reported stolen in March.
Det Supt Stevie Grant said: " I would appeal to anyone who has seen it either throughout the past few months or around the time of the incident to get in touch."
A major incident room had been set up at Kirkintilloch Police Office and extra patrols were being made to reassure the public.
The victim remains in hospital.
Ch Insp Craig Smith said: "Understandably people in the local community are shocked and alarmed at what happened and I believe a number of young children may have witnessed the incident.
"Police Scotland is working closely with our colleagues at East Dunbartonshire Council, and in particular the social work and education department, to ensure all those affected are provided with the best support possible."
Armed officers were called to the scene of the shooting, which happened as pupils were leaving St Helens at the end of the school day on Thursday.
One woman who lives in the residential area told BBC Scotland that she had heard three bangs which she believed were gunshots.
Several streets in the area were cordoned off, and access to the primary school was closed.

Iranian patrol boats surrounded a US cargo ship in the strait on Friday.
Earlier this week, Iranian naval ships reportedly fired warning shots near a Marshall Islands-flagged cargo ship before seizing it and its crew.
The strait connects the Arabian Sea and the Persian Gulf.
The US navy will not be escorting the ships - a different procedure from accompanying them. Instead, it will monitor the area as the ships pass across the strait.
The crew of the Marshall Islands-flagged container ship seized by Iran in the strait on Tuesday are safe and "in good spirits", Danish shipping company Maersk said.
Maersk said it still did not know the reason why the Marshall Islands-flagged vessel was seized, and noted it had been in an international shipping lane.
Iranian media cited an official as saying the ship was seized based on a court order in connection with a complaint made by a private Iranian company about debts.
The crew includes a British national, but no Americans, as initial reports suggested.
The narrow strait lies partly in Iranian waters.
Shipping traffic is allowed to pass through under an internationally-recognised protocol called "innocent passage", so long as the ships are not carrying weapons, collecting intelligence, or violating other restrictions.
Police investigating a shooting outside a primary school in East Dunbartonshire said a number of young children may have witnessed the incident.
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Defence officials have said that US Navy ships will accompany US-flagged commercial vessels through the Strait of Hormuz, to make sure they are not interfered with by Iran.