Bringing your own wagon/cart

I rarely ever can get my hands on a cart when I go to the bins. So instead I have to lug around a lightweight but large capacity bag, checkout when it’s full and super heavy, empty it into my car, then go back in for more. I have one of those outdoor wagon things and have seen them being used, so I was thinking about bringing it. But I’m wondering how that works when it comes time to checkout. Do they just believe you that it is yours and you didn’t just find it in the bins? Those things are heavy so I don’t want to have to pay for the weight of it each time obviously. How do you guys deal with your personal “carts” if you bring them with you?