Is it permissible to propose to a girl my friend wanted to marry, knowing that her family had initially approved of him then withdrew their approval due to his procrastination, and would I be sinful if a rift occurred between me and my friend because of this?

If your friend has not proposed to the girl, there is no harm in you proposing to her, for the Prophet, peace and blessings be upon him, said: "A man must not propose over the proposal of his brother unless the first suitor gives up, or gives him permission." However, if he has already proposed, it is not permissible to propose unless he is rejected, or he gives permission, or he gives up. Nevertheless, it is not chivalrous of you to go and preempt him in proposing to her, especially since he consulted you. Rather, you should wait until he gives up on her or her family rejects him. Strive to maintain the relationship, and seek guidance from Allah through Istikhara before proposing.