Should a girl agree to be engaged to a young man she met online, knowing that he appears committed and serious and wishes to propose to her soon?

The internet is not a means to discern people's morals and their sincerity in seeking marriage, because the person communicating may be deceitful. Therefore, it is advisable to cut off communication with this young man and ask him to formally propose to you from your family if he is sincere, while keeping previous communications a secret. If he proposes and you are assured of his religiosity, character, and competence, then seek guidance from Allah (perform Istikhara) and marry him if your family agrees. Otherwise, be patient and ask Allah to grant you a righteous husband. It would be preferable if you presented yourself, through your guardian or friends, to someone whose religiosity and character are approved.