If the head of the household refrains from offering the Uḍḥiya (sacrifice) without a valid reason, is it permissible for the wife to ask someone else to purchase and slaughter the Uḍḥiya on behalf of the family, and would this be sufficient?

The Udhiyah (sacrifice) is a Sunnah for both men and women equally. If a woman has the financial capacity, it is Sunnah for her to offer the sacrifice on behalf of herself and her household, especially if the head of the household refrains from performing this ritual. The Udhiyah is a collective Sunnah (Sunnah 'ala al-kifayah) if there are multiple members in the household; if one of them performs it, it suffices for everyone. Otherwise, it is an individual Sunnah (Sunnah 'ayn).