Is it permissible, in Islamic law, for a young man and a young woman who are not related to each other to communicate online, with the knowledge and approval of the girl's family, and with the intention of marriage after graduation, while observing the Sharia-compliant guidelines for conversation?

Your maternal aunt's daughter is a non-mahram to you. It is not permissible for you to speak with her except out of necessity and to the extent of that necessity, because a woman speaking to a non-mahram man is a means to temptation, especially if she is young. So be careful about communicating with her until you marry her, or عقد (perform the marriage contract with) her so you can speak to her without embarrassment. It is better not to rush into the engagement (khitbah) or the marriage contract (aqd) until the marriage itself is soon.