Generally, no, but in some cases an exception may be made with permission from the business owner.
Under the Americans with Disabilities Act, access rights are based on "reasonable accommodations." While the US Department of Justice has not issued any guidance one way or the other on this specific issue, you can apply the principle of "reasonableness" to any situation to see whether the ADA permits/requires it or not.
Let's do that here. We have competing interests.
1. The health department has regulations about live animals being near food which makes it illegal in most areas for restaurants and grocery stores to permit pets into their facilities. An exception is made for service animals under reasonable accommodation under the assumption that the risk is minimal.
2. The business has an interest in activities that cost them customers. The public "ick" factor for a dog at or near table level is significantly higher than for one on the floor under the table.
3. A person with a small dog may prefer to keep their dog in their lap, but it is not an absolute necessity.
Reasonable accommodation means that you take ALL interests into account when evaluating a specific situation, not just whether the individual with a disability thinks it is reasonable from their perspective, but whether it is reasonable for all. The first item on our list addresses an issue that falls under "direct threat," because the health department has stated there is a sanitary and health risk of food contamination if live animals are near food. An exception has traditionally been made for service animals based on the long history of guide dogs, who are always on the floor, tucked under tables, and as far away from the food as possible. The risk of food contamination when the dog is on the floor is very minimal. This is where normal pet owners have their pets during human mealtimes, if not completely out of the dining area. Putting a service dog in a lap is significantly pushing the envelope of safety on this issue.
The second item on the list is an example of "undue burden." While some customers may leave because of the mere presence of the service animal, it is not possible to accommodate the disabled patron at all without the service animal. That situation falls in the favor of the person with the service dog. However, in the case of having the dog close to the food, the balance shifts. The business will lose even more customers, but for something that is not strictly necessary. This is unreasonable because it creates an undue (avoidable and excessive) burden for the business person. However, if the business owner has no objection to the dog in a person's lap and is willing to do additional cleaning to satisfy health department regulations, then the business is permitted to make an exception IF THEY WISH. They don't have to.
Now we examine the third item, or an individual service dog owner's wish to have their service dog in their lap. This is not strictly necessary. A small service dog can sit or lie under the table just the same as a service dog of any other size. Their size alone does not make it necessary for them to be in a lap. It is sometimes claimed that this is a safety issue because smaller dogs are at greater risk on the floor than larger dogs. In fact, a small service dog is far safer under the table than is a larger service dog because he is less likely to have body parts extend out from under the table to get trampled. So there is no reason to treat small service dogs differently for safety.
Some owners will claim that their dog is not able to work from the floor. Since there is no legitimate job a small service dog can do that a larger one cannot also do, including "alerting," this is a problem of the owner's own creation. If they failed to train the dog to be able to work from the floor, that is their fault and within their power to correct. It is not reasonable to expect a restaurant to compensate for the handler's lack of foresight or poor choice in dog.
Most legitimate small breed service dogs are perfectly capable of working from the floor, including heeling and long stays, exactly the same as their larger counterparts. There are some limitations, including an increased risk of injury in crowded environments when it may be prudent to lift the dog from the floor and carry it until the crowd thins out. A smaller service dog will not be able to heel as long, as far, or as fast as a larger service dog can. This is an obvious flaw in their use as service dogs that the potential owner should consider carefully when making their choice of a service dog. If you choose a dog with limitations, then you are stuck with those limitations and you are responsible for dealing with the consequences of your choice. There is no task or disability that specifically requires that the dog be tiny. It is a personal preference, a personal choice, with consequences. A person is free to make that choice, but they are not free to inflict added burdens on others because of that choice.
Now we get to the issue of small dogs in shopping carts, pouches, or other carriers that hold the dog high off the floor while shopping in grocery stores. The purpose of the shopping cart is to carry groceries and small children. It is a courtesy provided by the store for its customers because shopping carts will encourage them to stay longer and purchase more. Carts are not there as a convenience to service dog owners. A public accommodation is not required to provide you with equipment to care for or utilize your service dog. If your dog is capable of working without a shopping cart in other venues, such as offices, stores, theaters, etc., then it is capable of doing so in a grocery store as well. If an individual store gives permission for the service dog to ride in the cart it is permitted, so long as it does not also violate health codes.
It is permissible for a service dog to be carried by hand or in a carrier about the owner's chest in SOME areas of a grocery store, such as areas with canned and commercially packaged foods displayed on shelves. It is not reasonable to hold a service dog above the edge of a display of fresh food (such as vegetables or meats) which are in a cooler or bin that you might lean over to reach the food. It is not reasonable to hold the dog above the edge of a buffet style display where customers serve themselves, such as a take-home salad or food bar. This would permit hair and dander to fall onto the food and contaminate it. A rule of thumb is that whatever height the dog is from the floor, that is the distance the dog should be away from any open display of fresh foods. If the dog is three and a half feet off the floor and on the handler's person, then the dog and handler should not get within three and a half feet of the open display. This will prevent hair and dander from being deposited on food in the open display.
By the same token, large dogs that are able to see over the edge of the open display should never be permitted to look over the edge of the display, and should be worked from the side of the handler that is furthest from the display, such that the handler is between the dog and the display.