You have to start OBS with a command line argument, I think it's --portable.
An alternate way of doing what you mentioned is to set the base resolution to your desktop+webcam, record both and then cut in post.
Example: Your webcam runs at 1280x720 and your desktop runs at 1920x1080. Set your base resolution to 3200x1080 (1280+1920 for the width, and the tallest source for the height - 1080). Then put both in your scene.