Before my son was diagnosed with Autism, he did in fact have pure meltdowns and screaming tantrums in stores. It was very difficult to manage him in the store. Eventually, he was diagnosed with Autism and sensory disorders. To this day, going to a store is very difficult for him even though he is older.
Some tips which I've learned that can seemingly help all children are:
1) If the child is having a tantrum in the store, figure out what's causing the tantrum. Is the child tired, hungry, overstimulated? Sometimes it's a matter of the child wanting to run loose in the store. That's always been a big no no for my family.
2) Starting from a young age, place the child in a cart while shopping at the store. The child will learn that that's where they sit and they are not allowed to run loose to prevent injury to themselves and others in the store. (There's nothing worse than when I'm walking in the store and a child darts out in front of me almost getting injured which could have been prevented if the parent didn't let them run loose.)
3) Don't bribe your child with rewards for bad behavior. It's ok to offer a reward to the child for a job well done listening while in the store. But if the child doesn't comply, don't reward the behavior.
4) Keep a chart for the child at home, age appropriate, which rewards the child with something for good behavior at the store.
5) Talk to your child and let them know what's expected of them at the store. Give them advanced notice so at the store the transitions will be easier.
6) If your child's behavior doesn't stop and it's too difficult to continue shopping, leave the store and go back later.
7) When my children were smaller, I packed a goody bag of stuff they could do in the cart while I did my shopping. Now they bring their electronic games to prevent boredom.