Wow, after reading the posts here I am sorely amazed and concerned at what is going to happen to some of the parent/child relationships once they hit the teen years! Some of the advice you've received is good, others is stunningly terrible!
I have a 16 yo daughter, and while she hasn't exhibited some of the behaviors your daughter has, I can tell you without a doubt that the majority of the reasons why teens do this is due to one of several things:
A desire to be independent, ie, not having to ask mom or other family member for EVERYTHING they get.
A peer pressure issue, "come on, its really no big deal" and bragging rights later to fit in, "You'll never guess where I got this..."
A subconscious way of asking for attention, after all, good or bad, at least she's getting it.
I am not alluding to the fact that you ignore your daughter, neglect her, etc. Just listing a few possibilities. Not too long ago, my teen daughter told me that she wouldn't want to do some of the things that other girls were doing because she valued our relationship w/o the drama over the "thrill" of fitting in. I am extraordinarily blessed to have a daughter who has realized this without having to directly point it out.
I would definitely have your daughter apologize to her grandmother, with sincerity! Be present, not to harp, but to lend support to your daughter. Grandma should accept the item back and not dismiss her apology, just as another mom suggested.
I would begin to develop a relationship with your daughter that signals a different phase in her life. I am not suggesting you change the home rules for your daughter, but a different level of relationship is called for. She is almost an adult and deserves a few more liberties than her younger siblings. Take her out to lunch, just the two of you, take her on shopping trips and give her her own money to spend (my daughter has to earn it but I pay her generously). Show her that the relationship between the two of you is far more special that the rupture caused by theft or other illegal activities.
While I think there can be a place for a child to tour a police station, etc. I think those are more tough love actions on a child that is teetering on becoming part of the population and not for a young lady who has made a bad choice. She doesn't need boot camp for stealing a purse for heaven's sake! I think there are a few mothers here who need to look with love in how they would deal with their own child who made some bad choices and remember your own! At least I hope that would temper some of the "advice" given here, or they'll have one heck of a problem on their hands.
Blessings and good luck to you and your daughter!
C.