Is it possible to issue a fatwa for divorce for the aforementioned woman, based on the absence of a dowry or gifts, and the matter being limited to a verbal marriage contract, "I have given you my daughter in marriage... I have accepted," with the husband refusing to divorce her?

If the legitimate marriage contract for your sister with this man was concluded with offer and acceptance, a guardian, and witnesses, then she has become his wife. It is not permissible for him to leave her in limbo; he must either divorce her or complete the marriage with kindness. If he does not divorce her, refer the matter to the competent authorities, such as Islamic centers, to compel him to do so. There is no way for her to marry another unless her husband divorces her or the judge annuls the marriage, for a judge's ruling is binding, unlike a mufti's fatwa.