Is it permissible for a son to financially assist his wife's mother in purchasing furniture without his own mother's knowledge, and is concealing this considered disobedience to her, warranting punishment?

Defining what parents must be obeyed in and what they must not is a difficult matter. Working for your own benefit, completing your marriage, and cooperating with your wife's family in this regard are desirable and good, as Allah Almighty says: "And cooperate in righteousness and piety." Pleasing one's parents is a religious obligation, for in the Hadith: "The pleasure of the Lord is in the pleasure of the parents." If it is possible to convince your mother of what you want to do and that it is beneficial for you, then do so. If you cannot do that, and the completion of the marriage depends on your assistance to your wife and her family, then the most likely course of action is to help them while avoiding your mother's knowledge of it, trying to deliver the money indirectly so that you can use dissimulation if you need to.