I have some recollection of a study I once came across (as a warning, it was somewhat disturbing) about the cognitive development of dogs, where the researchers took a number of puppies and somehow arranged things so that the puppies wouldn't see any changes in their vision while moving. Something like they were kept in a box with their heads secured facing one side of the box, with a treadmill that would occasionally run so that their muscles still developed. And what they found was that the puppies were rather severely non-functional when they were finally removed from their boxes. I think it was along the lines of functional blindness and a lack of normal canine volition to move around and do things, and the puppies weren't able to go on and develop normally afterwards. The conclusion was something about the importance of the visual field changing in response to their bodies trying to move around. Frankly it seems like a "no shit, Sherlock, was such an unpleasant experiment really necessary?" kind of conclusion, but it seems in line with your intuition about the crucial binding between vision and the motion of various limbs (and implicitly, that the environment actually produces different -- and to our visual networks, useful in some particular ways -- inputs as we move our limbs and neck around).

By the way, if anyone happens to know of the study I'm thinking of, I'd love to have a link. The last time I tried searching on Google / DDG, the results were drowned by unhelpful articles for dog articles.