What is the religious ruling regarding a young man who confessed to raping a five-year-old girl, and what should the girl's family do if the state does not implement the Sharia ruling, especially since they are secretly planning to kill him for fear of scandal?

If it is proven that the young man raped the girl, her family is not permitted to kill him. Killing is a grave matter and a major crime. Allah has threatened the intentional killer with Hell, and by doing so, they would be committing a greater evil than what the young man perpetrated. The implementation of hudud (prescribed punishments) is entrusted to the ruler, not to the general populace. The girl's family can raise the matter with the ruler, or with secular courts if they are able to restore rights and repel injustice.