One dreaded task I try to avoid is going to the store with all 3 of my kids mainly because it's hard to get them to call down with their sensory issues. The truth of the matter is there are days I HAVE to do it and slowly I am getting better. This is one thing I have learned is that it's amazing what happens when my children get to push the cart (especially when it's heavy by adding groceries or a sibling). I will stand at the front and steer it but they use their muscles and push. If there is a fight over the cart I make sure I get a basket for the other to carry and I fill it according to the child who is holding or dragging it across the grocery store. Sometimes they get tired but I try to make a game out of it. 'Can you see if you can push it to the end where you see the chips?' I try to make them think and use other ways to move the cart besides pushing with their hands on the handles.
Maybe if you are someone who as the toy shopping cart you can use that instead although a stroller or a basket with wheels will work just as well and fill it with heavy items and have them push it around the room.