Is supplication alone sufficient as a means for a woman who wishes to marry but has not been able to, or is it incumbent upon her family to build social relationships and search for a suitable person, especially since they are not in their home country and do not know anyone?

It is not obligatory for the family to establish social relationships for the purpose of marrying off their daughter. However, it is مطلوب (sought after) and مندوب إليه (recommended) for the guardian to strive to marry off his ward, and the best of people, like Shu'ayb and Umar, did so. If custom prevents this, some people can be intermediated to present the matter. It is permissible for the woman herself to propose marriage to someone she deems righteous, within specific guidelines. Supplication (Du'a) is one of the most beneficial means.