Service Dog Central

Legally there's no requirement that they must be permitted or that they cannot be permitted. Some local health codes may not permit it and some grocery stores may object to having them in the child seat just as some restaurants would object to having them on the table.

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.

They can't really perform any tasks from the child seat and aren't safe there, so there's not much point in putting them in a cart anyway.