Is it permissible to speak with someone with whom a vow was made upon the Qur'an not to speak to them unless through a legitimate marriage contract, knowing that the legitimate viewing will take place next week?

First, sincere repentance is obligatory. The suitor is a stranger to the fiancée, and it is not permissible for her to speak with him except out of necessity, with controls to prevent temptation. If she wishes to fulfill her oath, she should not speak to him before the legal marriage contract. If there is a necessity to speak with him under legitimate controls, then there is no harm, and she is obligated to offer an expiation for breaking an oath.