Is it permissible for a merchant to stipulate to his new partner that he not deal with his customers after the partnership ends, if the partner agrees to that?

The fundamental principle regarding conditions is permissibility and validity. Nothing is forbidden or invalidated among them except what the Shar' (Islamic law) indicates as forbidden or invalid, based on the saying of the Messenger of Allah, peace and blessings be upon him: "Muslims are bound by their conditions." This is also because the default assumption is the validity of contracts and conditions, as long as they do not contravene the Shar'. So, if the partner accepts this condition, he is obligated to observe it.