Is it permissible to take food from my married brother without his permission, knowing that he dislikes it, given that he is responsible for our sustenance but refuses to fulfill our requests?

If you believe your brother would dislike you taking something from his food without his permission, then it is not permissible for you to take it. However, if your maintenance is obligatory upon him, then it is permissible for you to take what you need in a reasonable manner without his permission, by analogy with the permissibility for a wife to take her maintenance from her husband's money if he withholds it from her.