Lost a jacket and my co-worker's sister has it, how do I talk to her about it without sounding accusatory?

Don't confront the coworker! She isn't the one in posession of it! You have seen her sister in it several times and you know the coworkers parents. Apparently, this must be somewhat of a small town. So confront the sister you have seen with the jacket and tell her you have pictures of it as proof. The coworker doesn't have anything to do with your coat missing, her sister does. You can't prove the coworker stole it from you at work can you? And leave personal issues at home, don't bring them to work.