Right after Webb runs out to get Eddie and finds him, he goes to shoot the attackers who are driving away and gets shot twice in the back, with Conner appearing above him shooting at the attackers. It's heavily inferred that Conner shot him, as he made a point of Webb wearing a vest but what was the point of it?
That always puzzled me. My understanding is that Conner didn't want this incident to escalate to becoming an international scandal with the Japanese men engaging in a shootout with police. Throughout the film, Conner seemed very concerned about keeping good relations with Mr. Yoshida and that Keiretsu (a group of Japanese companies working together to engage in negotiations with foreign countries but at the same time they rival one another). Eddie's dad is the head of a company that rivals the Nakamoto Corporation headed by Mr. Yoshida. Heading back to the topic, Conner realized that the Japanese's sole objective was to kill Eddie and once they achieved that objective then it should be left at that. That's my take on it.
I know in movies people are always crack shots when required by the story, but to risk fatally shooting your partner to save face for a 3rd party is nuts. I know he's Connery but I'd love the idea of an old guy, at night, deciding he could accurately hit my vest while we're both in motion. It seems a simple shout of "Webb, hold your fire" would have done the trick. Plus why would he bother shooting at them himself, even for show if his goal was to stop Webb from shooting at them? It's a part of the movie I've always gone WTF at.
A lot of things in the film just don't make sense and they were probably there just for style. It's not Connery's style to say: "Webb, don't shoot!!" but shooting him in the back instead.
I never assumed or inferred Connor shot him. I always thought he was shot by a sniper off screen working for the police or maybe the Japanese. We don't see John Connor draw down and shoot him, so why assume it?