Yes, it should move. It's the "lever" you move to put the machine in bobbin winding mode. Pushes to the left if memory serves. You'll see the whole bobbin case move upward when you push it to the left. Then to engage it in sewing mode, just close the slide plate. There are probably some spots to oil on it,.. I'd have to look closer as the one I have here waiting for a tuneup and feed dogs.
It should show an oil point in the manual, either from the top in the bobbin area, or from the bottom, I'd think. Either way, just oil anything where there's metal to metal contact, and don't get oil on those feed dogs! Oil is what ruins them the fastest.
It's not the end of the world, as long as it's in "Sewing mode". If you simply couldn't get it working to wind the bobbin, you could use a side winder. I think they are supposed to do a better job of winding a bobbin than a lot of the self winding mechanisms anyway, especially when they get out of adjustment.