I have set up a "community" site with K2, K2 Story, and Community Builder. The end user can write items/articles in the front end and save them and can create a list of their own items/articles for editing. To display the articles in a list, the articles are searched by category. If the end user did not enter a category when writing the item/category, the item/article will not be found. Is there a way to make the category field a required field so that the end user can not save the item/article without having entered a category first?

By default the category is required. Without a category cannot be saved.
If you are experiencing a different behaviour then it is possible that an extension is causing this.

I would start by looking into K2 Story's parameters.