How can I zoom without moving camera (also for render)? I tried a tracking scene and I want to zoom in, kinda like the viewport zoom but for my final render.

Viewport Zoom, I want it to look like this in my render
https://imgur.com/a/gjxhu0A

But when animating the camera to get closer, or changing the focal length the background of course doesn't move along.

Example for focal length/camera animated zoom, background doesn't move along.
https://imgur.com/Mpo7cJg

How can I do this?

(The gif file sizes were too big, I had to upload them elsewhere)
That is a background image, it is not supposed to move. It's only for reference. If you want the image to stay in place, while you move the camera use the addon "import image as plane". Or simply create a plane with an emission material and use your BG image as the texture.