What is the reward of a woman who sponsors an orphan while her husband will pay the expenses?

All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

There is no doubt that sponsoring an orphan is one of the greatest acts whereby one brings himself closer to Allah. Enough reward is mentioned in the hadith in which the Prophet, sallallaahu ‘alayhi wa sallam, said, "I and the one who sponsors his own orphan or other people's orphan will be in Paradise like these two" – and he gestured with his forefinger and middle finger, holding them apart. [Al-Bukhari and Muslim]

Even if your husband will pay the expenses on your behalf, regardless of whether that is from your own money or his money that he has transferred to your possession, you will get the reward of spending on an orphan.

If you mean that you just want to guide your husband to sponsor an orphan, then, undoubtedly, you will get a similar reward. It was narrated on the authority of Abu Mas‘ood Al-Ansari that the Messenger of Allah, sallallaahu ‘alayhi wa sallam, said, “Whoever guides someone to do something good will get a similar reward.” [Muslim] 

Allah knows best.