I’m flagging John as lawful because he seems to really go in for the whole ‘queen and country, law and order’ dealio. He clings not only to justice, but also to propriety. Social structure is his bread and butter. He’s got every marker of a classic lawful character.
You could take his carrying a gun and shooting somebody as chaotic vigilantism. (Actually John’s killed two people on-screen; remember TBB when he kicked the crossbow so it shot Sherlock’s strangler instead of Sarah?) But the definitions of the alignments have nothing to do with a character’s willingness to act in support of their personal beliefs. After all, a paladin traditionally runs around smiting evil-doers. An actively lawful good person would reasonably act to uphold their morality in the absence of someone else who’s legally appointed to do so. E.g. John called the cops, but he knew they wouldn’t get there in time, so it was him or nobody. And leaving Sherlock to be coerced into taking poison because John refused to act would not be the action of a good man either, would it?
So I think John carries a gun not because he’s a natural law-breaker, but because he’s unwilling to cede his capacity to act just because he’s moved from a military arena where he wasa lawfully appointed authority to a civilian one where he is not.
One thing about alignments, though, is that they’re not static things. People do change over time. Under the influence of Sherlock, John may begin to sway more chaotic. Alternatively, he may remain lawful good but begin to replace the societal concepts of justice and rightness he previously oriented on with Sherlock’s interpretation.