Is it permissible for a young man and a girl to continue their relationship online after the young man marries another woman, with the intention of marrying her (the girl) in the future, if his wife refuses to annul the engagement?

A woman's conversation with a non-mahram man is impermissible except when there is a need, and only to the extent of that need. This is due to the strong suspicion of temptation (fitna) it carries, and because it is a gateway to evil. Anything that leads to temptation is impermissible. The desire for marriage does not make such conversation permissible; rather, a Muslim must be cautious for the sake of their religion and honor. It is permissible for a woman to seek a husband, to seek help from trustworthy individuals, and to offer herself to someone she wishes to marry, as long as she does not overstep the bounds of Islamic law.