Is it possible to animate zooming in on an object, or is moving the object the only option available to get it to appear closer to the camera?
You can animate Focal length in camera properties panel
Go to the first frame of you animation and insert a keyframe on the focal length, right click over the focal length and select Insert Keyframe value or press I)
Move to a different frame, change the focal length value and insert another keyframe.
You can animate the camera, just as any object. For instance, it's perfectly possible to animate the focal length:
You can also animate the camera position:
As you can see, the zooming effects of these two options are different, and which one to use depends on what you want it to look like. You could also animate both position and focal length together, to make it look like zooming in while moving away or zooming out while moving closer.
Moving and keyframing has already been covered here but there a couple of other things that can help make this very easy to work with.
To start with, if you select the camera and press the Number Pad 0, the viewport display switches to the camera's view point. You can also access this feature using the menu shown in the following image.
For now, if you move anything then the view will return to the normal viewing but in Orthographic mode, however! if you open the numeric panel using n or the tiny little plus tab that sticks out at the side of the viewport, you should see an entry named "Lock Camera To View".
When you have the camera selected and you activate the checkbox, you should now be able to move the scene around from the camera's point of view just as you would normally do if the camera wasn't selected, only now it's actually the camera that is moving.
It's very intuitive to use and works flawlessly for aligning the camera to your scene. When you exit that mode by deactivating the checkbox and you move the view again, the normal viewing should return except that maybe it will be in orthographic mode which has no 3D perspective.
You can return to the 3D view by pressing NumberPad5 or by using the menu shown above where it says "View Persp/Ortho"
And the customary animated .gif.(Unfortunately I had to go fast because I kept exceeding the 2MB limit for file size uploads.)
If you find that you can only rotate the view but you can't really pan around very quickly then you need to reset the view. This happens often with Viewport navigation and can be fixed with either NumPad . which zooms to the selected object or Shift+c which zooms to the entire scene. That happened to me when I recorded the .gif.