You’re not wrong or anything, but “on accident” is used commonly in American English, so the author isn’t wrong either. I think it might have come from an association with “on purpose”, as in “I didn’t do it on purpose, I did it on accident.”
Unfortunately (or fortunately, depending on how you look at it), the rules of language only matter if people actually stick to them. Language shifts over time no matter who kicks or screams about it.