What must I do after having vowed to fast on Mondays and Thursdays for my sister and her husband to reconcile, and am I sinful for not seeking my husband's permission, given my inability to fulfill the vow?

If you are continuously unable to fulfill a vow to fast on Mondays and Thursdays, then you owe an expiation for breaking an oath (kaffarat yamin). Do you need to feed a poor person for each day? There are two narrations in the Hanbali school of thought: the more authoritative one states that you must feed a poor person for each day, while the other narration states that nothing else is required of you besides the expiation for breaking an oath.

However, if your inability is not continuous, and your husband has given you permission, then you are obligated to fulfill the vow and fast the vowed days. If you break some of the fasts, you must make them up. If your husband does not give you permission, he has the right to prevent you from fasting, and you must make up the days you missed due to his prevention, and fast on the days he does not prevent you.