Restaurant owners on the South Fork say there has been a recent proliferation of customers seeking to bring pets to the dinner table with them, even in some of the region’s most elegant dining rooms—with a growing number of diners seeking to take advantage of a status created to ease stress about airline travel.Health codes do not allow animals in establishments that serve, sell or prepare food. When informed of that rule, some customers argue, citing the fact that the Americans With Disabilities Act allows service dogs in restaurants and other food establishments. Many will then produce a letter stating that a beloved pooch is an “emotional support animal” and must be treated like a service dog. “It has been very difficult for us,” said Millie Fellingham, owner of Fellingham’s in Southampton Village. “I am an animal lover, but people push everything too far, you know? “We had one family, the kids came in and said their mother needed her animals with her. But then she has it on her lap, and its head is on the table. It’s been a problem on our patio, which seems like it’s outside, but it’s not.”At Cowfish, in Hampton Bays, manager Pat Jones said that the restaurant has had to tighten its policy regarding dogs on the property because of the rising number of four-legged patrons being toted through the doors. “We are trying to encourage people to leave their dogs at home,” Ms. Jones said. “If someone comes with the proper information for a service dog, they may sit on the patio. And we have an outdoor bar, where someone can sit and have their dog with them, as long as it is not on their lap.” Even at East Hampton’s celebrated Nick & Toni’s, owner Mark Smith says, the restaurant has had to contend with customers wanting to bring dogs into the posh restaurant. “The problem is, there are some people who, at some level, the dog does help them, but there’s always people who will take advantage of that,” Mr. Smith said. At the company’s casual burger joint, Rowdy Hall, a sign next to the front door reads, in bold letters: “We Love Dogs.” But it then goes on to explain that the Suffolk County Department of Health allows only certified service dogs into restaurants—and that “emotional support dogs” do not qualify. “We ended up having to just get to the point,” Mr. Smith said. Some restaurant owners and managers have said they have encountered combative customers who insist that the restaurant cannot make them identify a handicap or show certification that the dog is a service dog and not just an emotional support animal. But the actual rules regarding animals in restaurants and other food establishments are cut and dried: In Suffolk County, only a trained service dog is allowed in a restaurant, as they are nationwide in accordance with the Americans With Disabilities Act. There is no letter from a doctor or therapist that qualifies a dog, or any other animal, as a service animal. To be a service animal, the dog—only dogs are eligible—must be trained to perform a specific task in service of its owner. There are indeed emotional and psychological afflictions for which service animals can be trained. For instance, people who suffer from seizures or post-traumatic stress disorder often have dogs trained to detect symptoms of onset and protect or comfort their owner when there is an issue. “We follow the ADA, in that the sanitary code allows service animals in food establishments,” said Stephen Kane, the assistant bureau chief of public health protection for the Suffolk County Department of Health. “It’s a dog, no other animal, that conducts a service for a person with a handicap. Beyond that, no animal has to be allowed access.” A common claim by animal owners is that an establishment may not ask for any kind of proof that the animal is a service dog. That is only partially true. The ADA, according to Mr. Kane, allows any establishment owner to make two inquiries for evidence that the dog is an actual service animal: The owner may ask if the animal is a service dog required because of a clinical disability, and what the service or task is that the dog is specifically trained to perform. It is true that they may not demand documentation, an explanation of the person’s disability or a demonstration of the task the animal is trained to perform. Nonetheless, emotional support animals are a growing breed, or breeds. Literally dozens of websites offer emotional support animal “certification” for dogs, cats, birds, pigs, guinea pigs and other animals. Some, like one calling itself the Emotional Support Animal Registration of America, allows a pet owner to self-register a dog, for a fee, and says falsely that owners will be able to “bring your animal anywhere in public—it is permitted.” Others sell emotional support animal vests designed to resemble the sort of high-visibility jackets that trained service dogs typically wear. The rush for emotional support animal certifications has mushroomed largely because of airline policies. The term itself is one coined by therapists to identify the once-rare case where a person who suffers from anxiety over flying is comforted by having a pet—which not too long ago would have had to travel in a cargo hold—on their lap during the flight. As airlines have ramped up fees for flying with dogs or other pets—typically $75 to $100 each way, per animal—that are waived for emotional support animals, the number of such dogs in the main cabin of flights has ballooned. Airlines have started cracking down, requiring letters from certified therapists, and that dogs be well behaved regardless of a passenger’s need. But lawyers for airlines, and now other institutions like public buildings and even apartment building tenant boards, have determined that it is better not to take a strong stand against the burgeoning number of “certified” animals, for fear of risking a discrimination suit from someone with a legitimate need.And psychotherapists say that there are plenty of cases in today’s anxiety-ridden society that wholly justify the minor matter of having a dog on someone’s lap. The use of a dog for emotional comfort, especially on airplanes, can be a welcome replacement for alcohol and prescription drugs for treating anxiety over flying. “For people who are afraid to fly, I think having a dog with them is a wonderful way of helping them instead of drinking and taking a bunch of Xanax,” East Hampton psychotherapist Mary Bromley said. “There are people who suffer from panic attacks for whom having a small animal with them is extremely comforting. But it’s really only for airlines—that’s it.”But Ms. Bromley—who said she writes about four or five letters a year in support of one of her patients being allowed to have an emotional support animal with them and turns away many more who request the letters just to avoid airline fees—also said that for some people with severe anxiety issues having a pet at the dinner table with them in a public place would be a perfectly reasonable solution to keep them calm.“I’ve only recently become aware that people are using those letters at restaurants, and there are times when that is appropriate, too,” she said. “I would say that is okay with me in those specific cases of people who have panic attacks or extreme anxiety or PTSD from a rape or assault.” In 2015, New York State adopted what is known as the “Doggy Dining Law,” allowing dogs in outdoor seating and dining areas—if the restaurant chooses to allow them—with certain restrictions on the dog’s behavior and what they can’t do, like being fed off the table. Most local restaurants seem to have adopted the approach that they can keep most of their customers happy and satisfy the requirements of law by allowing any and all pets in their al fresco dining areas. “The last thing we want to do is make someone uncomfortable at our restaurant, and the other last thing we want to do is make someone at the table next to them uncomfortable,” said David Lowenberg, who is a co-owner of Bell & Anchor in Noyac, the Beacon in Sag Harbor and Fresno in East Hampton. “We welcome guests with their pets in our outdoor dining and simply ask that they be cognizant of other diners.” Other establishments go out of their way to make sure their customers know that dogs are welcome, within certain confines. The Maidstone in East Hampton has been listed on several online lists of “pet friendly” establishments in the Hamptons, and its staff does everything but roll out the red carpet for pets and their owners. “We are dog friendly,” general manager Ben Levine said. “We have outdoor and public spaces where pets are always welcome, though never in our main dining room.”He added, “We like dogs—more than people sometimes.”