Which motherboard do you have? If you populate that PCIe slot are you going to be stealing lanes from the others?

The only thing that comes to mind is one of the BlackMagic capture devices:

https://www.blackmagicdesign.com/products/decklink
https://www.blackmagicdesign.com/products/intensity

But I'm pretty sure those solutions are going to require you output from your main machine into a secondary machine which is dedicated to streaming.