Is it permissible for a woman to go to the market despite her husband's oath of divorce and prohibition upon her for not eating with him, especially since he constantly neglects her and their children, and only with difficulty buys them their necessities?

Your husband commits grave sins, treats you poorly, and neglects God's rights and your financial support. In such a case, separation from him is recommended. If he persists in his behavior, it is better for you to seek a divorce or khulʿ. As for his conditional oath of divorce and tahrim, if you go to the market, the divorce and tahrim would take effect. However, since he does not provide for you, you are permitted to go out without his permission, but doing so would break the conditional oath.