Is there a religious reward for spending from one's private money on household needs if the husband is financially capable but does not provide the money, and is this spending considered charity?

If the charity is voluntary and not zakat, then it is permissible for a woman to intend to give charity to her husband and household when purchasing household necessities. She will be rewarded for this, even if the husband is not poor, because voluntary charity is permissible for the wealthy. Spending on the husband and children combines the reward of charity with the reward of maintaining kinship ties. It must be noted that the wife is not legally responsible for expenses; rather, all financial responsibility falls on the man. If a woman intends to give charity and draw closer to Allah Almighty, it is not permissible for her to take compensation for what she has spent.