I'm going to tell my niece off in front of the family and have them turn on her
Why would you do that to your niece?
She had a bit too much to drink and she threw up on the family carpet
It's not okay to try to instigate a problem between your niece and other family members. Instead you should talk to your niece about the dangers of drinking too much. If you talk to your niece, you will gain her respect, instead of her distrust.
I think I'm going to do it anyway, because I can.
I really don't think you should tell of your niece and try to get the rest of your family to dislike her. It seems very vindictive, and not something that a mature person would do.