Parenting Teenagers: The Tattoo Debate
I am not sure about the laws in each state, so you may have to check to see at what age kids are allowed to get a tattoo without parental permission. My advice, however, is that as long as your kids are under 18, they are still minors and you get to make the final decisions. And many parents decide that their rules stand as long as the kids live under their roof.
Talk to your daughter about why she wants a tattoo and explain to her why you don't think it's a good idea. Have a discussion, not an argument. Then let her know what your rules are and when she will be able to make that decision for herself.