The short answer is: No, businesses do not have to permit dogs, including service dogs, in the shopping carts owned by the business.
This is partly an issue of fundamental alteration and partly one of reasonableness. It is also an issue of professionalism and image.
Public accommodations must ... make reasonable modifications in policies, practices, and procedures that deny equal access to individuals with disabilities, unless a fundamental alteration would result in the nature of the goods and services provided.
The fundamental purpose of the cart is to transport merchandise, not dogs. Co-opting the cart from that purpose to transport a dog changes the nature of the service being provided (the cart).
In addition, the the business is not required to provide services or equipment for the dog, just to permit the dog to accompany the disabled owner the dog is trained to assist.
9. Q: Am I responsible for the animal while the person with a disability is in my business?
A: No. The care or supervision of a service animal is solely the responsibility of his or her owner. You are not required to provide care or food or a special location for the animal.
Except for airports, businesses are not required to provide toileting facilities for dogs either. They don't have to provide food, water, or dishes for the dog to eat or drink out of. If they aren't required to do any of that for service dogs in general, why would they be required to provide equipment to transport the dog while in their business?
Consider the scooters, power chairs, electric or manual wheelchairs or other mobility devices many large stores provide for customer use. They are not required to do that. It is a courtesy some stores choose to provide as part of their customer service to make the store more appealing to more potential customers. They aren't required to provide transportation equipment for the human with a disability, and neither are they required to provide it for the person's dog.
Remember that public access laws are about access for the person with a disability. Whether or not the animal is permitted in the cart does not affect the person's access to goods and services. It therefore becomes an issue of reasonableness. The person with the service animal must have some way to manage their dog when in a public accommodation that does not offer shopping carts. (Some examples of accommodations that do not offer shopping carts include restaurants, offices, hospitals, movie theaters, and hotels.) Therefore there must be some reasonable alternative to putting the dog in the shopping cart if the store objects.
If a business gives permission for the individual to use the carts to transport their dog while inside the business, that's between that business and that team. The business can give that permission if they choose to do so but they are not required to do so. Some local health codes may not permit it and some grocery stores or customers in grocery stores may object to having them in the child seat just as some restaurants would object to having them on the table. You can argue whether a dog is different from a diapered child with the business if you like but you still need their permission.
There are two remaining considerations.
First, putting a dog in a cart raises the dog higher than even a traditional sized dog would stand on the floor. This substantially increases the range where hair and dander from the dog can be deposited. Here is a basic formula for calculating where falling things that do not drop straight to the ground may be deposited. Measure the highest point of the dog from the ground. Now in your head mark a point directly below that highest point and inscribe it with a circle of a radius equal to the dog's height from the ground. For example, if the dog's ear tip is three feet above the floor, then hair and dander can be deposited any where within a cone with a three foot radius at the base (on the ground) and a tip at the tip of the dog's ear. In order to prevent hair and dander from falling on fresh food, refrigerated or frozen food (which attract hair and dander), or ready to eat food, the dog must be positioned at all times so that that imaginary cone does not come in contact with those types of food. Raising a dog off the floor increases the bubble you need to maintain between that dog and food.
Second, carrying a dog or putting it in a shopping cart is not professional. It brings into question the legitimacy of the dog as a service dog because so many pet owners treat their dogs like fashion accessories. It also increases skepticism for other kinds of service dogs when businesses start seeing an increase in the number of dogs being brought in that really don't look like service dogs. It makes life especially difficult for those people with legitimate service dogs that happen to be small and significantly increases the likelihood and frequency of access issues for the person putting their dog in the cart.
Special considerations for owners of small or large service dogs
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?