If you must run two PC's, one for broadcasting, without a capture card then you can do it. By sending the OBS capture un-encoded over ethernet, then "capturing" that on the broadcasting PC, which then encodes and sends it out. The thing is that it wouldn't reduce the strain much on the gaming PC, which is the whole point of a separate stream PC. Some posts about it https://obsproject.com/forum/threads/encoding-over-lan-and-gpu-assisted-encoding.329/ https://obsproject.com/forum/threads/separate-computer-for-encoding.5064/
Capture cards really aren't very expensive. Like $100 max is plenty for Twitch, as often you don't even stream above 720p/30fps, which doesn't require a beast of a capture card. Example...