Is it permissible to send the monthly sponsorship amount to the orphan from the husband's salary, while still receiving the reward for sponsorship, in the event that this is not possible from the wife's salary?

It is permissible to send sponsorship money from the husband's wealth. If the husband donates the money to his wife, she receives the reward for sending it. If he appoints her as an agent or sends it with her consultation, she receives the reward for guiding to good and for acting as an agent therein.

As for the wife sending sponsorship money from her husband's wealth without his knowledge: if he customarily permits it and the amount is small, it is permissible, and she receives the reward, and he receives the reward for earning it. If he does not customarily permit it, or the amount is large, then it is not permissible except with his explicit permission. Customary permission takes the place of actual permission, unless the husband explicitly forbids it.