What is the legitimate Islamic solution for helping an unmarried sister, and what can be said to a religious young man to introduce her to him?

There is no harm in you proposing marriage to someone you wish to be your sister's husband. It is important to seek out a person of good religious commitment and character. Indeed, Umar ibn al-Khattab proposed his daughter Hafsa to Uthman ibn Affan, and the righteous man proposed one of his daughters to Musa (peace be upon him). You can also seek help through prayer to Allah Almighty, asking Him to facilitate a righteous husband for your sister.