Sorry, I've never used an Osprey card, they're not exactly common among hobby streamers. The web page doesn't say anything about being able to use all four inputs simultaneously, are you sure it's able to do this?
The web page also mentions an application called SimulStream, if this is a virtual DirectShow filter used to present all four video streams to other applications, you will most likely have to install it, and it's possible that you'll have to use the 32-bit version of OBS for it to show up in the device list.