Is it permissible for my brother to go with his brother's wife (my maternal aunt's daughter) to the market alone? And is it permissible if one of her children accompanies them?

A woman merely going out with her husband's brother to the market is not blameworthy, as long as there is no fear of temptation and no forbidden religious matter results from it, such as seclusion (khalwa). However, if there is no need for it, caution is obligatory, especially with repetition, for blocking the means to the forbidden is a legitimate legal principle.

The husband's brother (al-hamu) is dangerous, as the Prophet, peace and blessings be upon him, said: "The in-law is death." This is because evil and temptation from him are more likely due to his ability to access and seclude himself with the woman. He is more deserving of being prevented than a stranger.

You have done well by disapproving of their actions, so continue to advise them kindly. You can use avoidance (hajr) if you hope it will benefit them; otherwise, leave them and make abundant supplications for them.