Is the family considered sinful for refusing the sister's request to leave work, which led to her death in an accident?

You bear no sin for refusing to let your sister work, as that refusal was neither a direct nor an indirect cause of the accident that befell her. What is most important is to supplicate for her, seek forgiveness for her, and give charity on her behalf, for these acts are more beneficial to her than preoccupation with regret and self-reproach.