The short answer: It is up to the individual business to decide whether or not to permit their shopping carts to be used for transporting dogs, including service dogs. They are not required to permit it and in some cases might not be able to permit it even if they wanted to due to local health codes.
Must stores permit a service dog to ride in a shopping cart?
Is it okay to have a small service dog in your lap in a restaurant or in a pouch/purse/carrier in a restaurant or grocery store?
Special considerations for owners of small or large service dogs