No harm no foul. It's not like you could have known prior. If you wanted to establish just a shady-but-well-to-do background, the Rogue's Guild in Limsa Lominsa is basically... Well, they're a legit establishment of Robin Hoods.
It wouldn't impact your background much at all if they simply got taken in by one of the Rogues. You can access the quest My First Daggers once you unlock the armory system and reach level 10. Then it's just a visit to Limsa Lominsa and you're all set to start with the sneakysneaky.
It wouldn't impact your background much at all if they simply got taken in by one of the Rogues. You can access the quest My First Daggers once you unlock the armory system and reach level 10. Then it's just a visit to Limsa Lominsa and you're all set to start with the sneakysneaky.