In English, "have" is a very flexible word that can be used in a variety of contexts. I can have freckles on my face, I can have cake for dessert, and I can have a dead battery in my car. Furthermore, we can have parties, conversations, naps, arguments, elections, and children.
Getting back to your question, we can also have a taste of the soup.
There's only a small difference in meaning between "let my sister taste it," and "had my sister taste it," although "let" carries a connotation of allow or permit, while "had" might imply a little insistence on your part. In other words, had she asked for a taste, and you said yes, then "let" might be a better word to use. However, if you weren't sure the soup tasted satisfactory, so you wanted your sister to give a second opinion, then "had" might be a better word.
My sister thought the soup smelled delicious, so I let her taste it.
I thought the soup tasted funny, so I had my sister taste it.
Incidentally, yet another way to word this would be:
After cooking, I let my sister have a taste.
which combines both of those words, and would sound perfectly natural to a native speaker.