Has there been any woman mentioned by name in the Bible who we know to have died childless according to the text? If so, who was the first (by order of books)?

There are at least seven women in the Bible who were initially childless (also referred to as barren). Six of these seven finally did have children after several years. In Old Testament times it was a shame for a wife not to bear children (be childless) for her husband. Although many times it was the female in the marriage who was physically incapable of bearing kids, there were no doubt instances where the male couldn't engender children.

2 Samuel 6:22-23
New International Version
22 I will become even more undignified than this, and I will be humiliated in my own eyes. But by these slave girls you spoke of, I will be held in honor."
23 And Michal daughter of Saul had no children to the day of her death.