I want to create an application (C++) that contains a live stream to my webcam that is connected to the same computer.
I know that there is a way to realize this using DirectX, but my application needs to be platform-independent, so I think OpenGL would be better.
OpenGL is a library for drawing graphics, it has no means to read images from a webcam. I suggest you look for some open-source webcam libraries (there should be some, I imagine).
you should check out openframeworks. i am sure you can rip out a lot concering framegrabbing. it uses directshow or quicktime on windows, quicktime for mac, video4linux on linux (via compile-time-switches)