“It’s really hard to draw a bright line,” said Todd Weiler, a state senator in Utah who said that an old high school classmate of his keeps an emotional support pig. “To a large extent, everybody could benefit from having a pet,” Mr. Weiler said. “When is it an emotional support animal and when it is a pet?”

Sam Killebrew, a Florida state lawmaker who sponsored a bill to curb emotional support animal claims, said he went online and registered Ophelia, a stuffed baboon in his office, as his “emotional support animal,” even though she’s been frozen, her fang-filled mouth agape, by a taxidermist.

“As long as you pay your money, you’re going to get that card,” he said.

He sponsored a bill this year that would allow landlords to require that tenants who claim a need for an animal obtain a letter from a licensed medical professional based in Florida. Mr. Killibrew later withdrew the bill, but he said he planned to reintroduce it next year.

Sara Pratt, former assistant secretary for fair housing at HUD, agreed that the certificates sold online can be a problem. “They are useless,” she said. But she warned that state lawmakers who rush to criminalize people for seeking documentation of a need for a support animal are sending the wrong message to landlords, who are at risk of getting slapped with hefty federal fines.

The Americans with Disabilities Act defines service animals as dogs or small horses that are trained to perform specialized tasks, like leading a blind person or detecting seizures. Service animals must be allowed in restaurants, stores and other public places, even where animals are otherwise barred.

Emotional support animals, which provide comfort with their presence but generally have no special training, do not have the same status under the disabilities act. But when it comes to keeping an animal at home in a rental unit, federal law has been interpreted to give tenants a right to live with an animal if it helps treat depression or anxiety.