So, I have been to the supermarket several times since I received my Vivosmart HR, and the Vivosmart does not count my steps when I am pushing the shopping cart. Is this proper, and if so, how can I get the Vivosmart to count these steps?
A wrist-worn activity tracker can only try to guess at how many steps you've taken by way of wrist movements it detects. If you keep your wrists still (such as by resting your hands on a cart's handlebar while you push it), then it won't be able to detect steps you've taken.
Try pushing the cart only with the hand that is not wearing the watch, and leave the other arm to swing naturally by your side.