Can children be harmed as a result of breaking an oath, such as the statement: "May my children perish if I do not inform her," and does this oath require expiation?

The woman who relayed your mother-in-law's words was wrong and caused discord between you through forbidden backbiting. You should not have listened to her, but rather rebuked her. You erred by breaking your promise not to inform your mother-in-law about what the woman said, and you do not owe any expiation except repentance to Allah. Do not fear for your children because of curses invoked upon them; such expressions are considered vain utterances (laghw al-yameen) that carry no weight, for which Allah does not hold one accountable, and they are like a man's angry words to his family and children wishing them ill.