You don't punish? Then you're not properly preparing your daughter for the world.
Everyone who is employed, regardless of the status of their job, has standards in place, and clearly established punishments. A health care worker (whether she's a neurosurgeon or a medical assistant) must follow sanitation and hygiene practices. And there are protocols in place if the practices aren't followed (demoted, fired, counseled, pay withheld, suspended, arrested - depending on the severity of the violation). Every military member and emergency first responder has to adhere to rules, and punishment will follow if the rules are violated.
Every student, whether in first grade or in a PhD program, has standards to meet and rules to follow, and there are punishments for not following them.
Punishment is a consequence, a removal of some benefit or privilege, and punishment varies according to age and according to the violation. Cheating on the SAT test or plagiarizing on an exam? You fail. Talking in line while waiting to go into the museum on a field trip? You get a reminder from the teacher and the teacher or aide might stand right next to you instead of allowing you to stand freely like the rest of the kids. Punishment is logical. You leave your bike in the driveway after being taught how to put it away properly? You don't lose computer time, you lose bike privileges. You don't do your chores? You lose free time or allowance.
Punishment must be established before any rules are broken, after the rules are clearly taught, explained, and understood. Can you imagine if you were an employee of a business, and you broke an important rule, and the boss said "um, let's see, how about you don't get paid this week"?
You should explain calmly to your daughter about the expectations and the consequences. "We're going to the store. You will behave, you will not raise your voice, you will be polite to other customers and to the store cashiers and clerks". Then you lay out the consequences. "If you can't behave yourself in the store, you will not be able to play with your friends because you can't control yourself. You will stay home and you won't have electronics or tv."
It's not "lawyering". It's rude behavior. She needs clear established logical boundaries, after being quietly and calmly taught what the expectations and rules are.
And make sure she's getting rest, good sleep, and healthy nutrition. No energy drinks, no sugary sodas. Plenty of water, real food (not stuff out of the toaster or microwave), vegetables, etc. Get her a physical checkup and have her checked for vitamin deficiencies (B, D, etc and iron). A healthy body can contribute to calm behavior.