I hate hijacking threads, but it was either make my own for the exact same issue or just add a comment here.
I have the same issue. Only the active scene will display the webcam, and only after restarting OBS. The fix above by Digoot only works if you make that single webcam scene active BEFORE closing OBS, or the next time you open it you'll face the same issue again. It's a real pain in the butt to always switch to the webcam scene every time we close the program. We shouldn't need to do that. This is a bug.
So to summarize:
Say I have 3 scenes. Two of them have the same video capture source, the other has some other footage. If I open the program and it's on one of the scenes that has the video capture, it will display that webcam fine. If I switch to the other, it will NOT display the webcam. If I restart the program, the reverse happens. If I start the program on the scene that did not have a webcam, then the first scene I switch to that does have a webcam will be the only one to display it.
I have had SOME luck with either deleting and readding the video capture device scene, and I've had SOME luck with clicking "deactivate" and "activate" repeatedly on the scene I want the webcam to show up in.
Obviously while streaming or recording, this is so far less than ideal, I had to make a report about it here. Sometimes restarting the program just to switch scenes is not an option. In fact, in most cases it's not an option.
EDIT: Unlike EBrito mentioning transitions above.... changing the transition effect between switching scenes has zero effect.