Except Great Saiyaman would be classed a vigilante: no hero training, no hero licence and not working for any agency means he'd be attacked by real heroes……….Not that it would matter since Gohan could obliterate most of the heroes in his path.

If Gohan lived in a world where you could register to be a hero I don't see why he'd be a vigilante. Not sure what Chi Chi would think of it, on one hand it's an actual respectable career and it'd be highly unlikely there would be villains who could threaten him, on the other there's always the chance he'd be the one villain who'd actually turn out to be a threat to her son.