Does a divorce oath take effect if the wife authorizes her sister to buy necessities for her and her children without her husband's knowledge, given that he swore an oath of divorce if she were to do so herself?

If the husband intended, by preventing you from buying yourself, not to prevent you from buying by proxy, then he does not break his oath by you authorizing your sister. However, if he intended an absolute prohibition, then he breaks his oath by that.

Ibn Qudamah said: "In summary, whoever swears not to do something and then authorizes someone else to do it, breaks his oath, unless he intended to perform it himself."

If his intention in the oath is not clear, the fundamental principle is to consider his intention. If he had no intention, then the reason for the oath is considered.

If the husband breaks his oath, the jurisprudential opinion held by us is that the divorce occurs, whether he intended to effect it or merely to threaten. This is the view of the majority of scholars, including the four Imams.

If his divorce does not complete the three divorces, he can revoke it before the 'iddah (waiting period) expires.

Some scholars, such as Sheikh al-Islam Ibn Taymiyyah, held the view that whoever swears by divorce with the intention of threatening does not incur divorce, but rather owes an expiation for breaking an oath.

It is best to present these issues directly to scholars known for their knowledge and piety.