Is a husband allowed to ask his wife to search for a second wife for him, and is she obligated to obey him in that, knowing that his desire for polygyny is not due to any shortcoming on her part, but rather to his personal desire, financial capability, and ability to be just?

A wife is not obligated to obey her husband in his search for another wife, because the obligatory obedience is specific to matters of marriage and its related aspects, and it must be within what is good and permissible. The husband must fear Allah, treat his wife with kindness, and consider the natural jealousy with which a woman is created.