What is the Islamic ruling concerning an orphaned girl who became pregnant and gave birth twice by a young man without marriage, and now wishes to repent?

You must initiate repentance, distance yourself from this person, and abstain from this act. Be glad for Allah's love and forgiveness, for repentance erases what came before it, and one who repents from sin is like one who never sinned. Allah transforms evil deeds into good deeds for those who repent, as stated in His Glorious Words: "Except for those who repent, believe and do righteous deeds. For them Allah will replace their evil deeds with good. And ever is Allah Forgiving and Merciful." The best of days is the day of repentance, so praise Allah Who guided and enabled you. If you both genuinely repent, there is no harm in him entering into a valid, lawful marriage contract with you. As for your children born out of wedlock, they are not attributed to the fornicator, and you are responsible for their upbringing.