Mr. Haag said he was barred from a flight with his dog because airline workers did not believe it was a legitimate service animal. He said that ill-behaved emotional support animals, which have not undergone the rigorous training that service dogs receive, “make it harder for us to gain access” with dogs when a veteran may not have visible disabilities.

“My dog is specifically task-trained to help with flashbacks,” Mr. Haag said. “I do not have any physical injuries that you can see. It does make it more difficult to say what is my dog is for.”

Over 80 veterans and disability groups recently wrote to Elaine Chao, the secretary of transportation, seeking new rules that would require that emotional support animals be trained if allowed on planes. A number of websites offer “certification” for emotional support animals, for a fee. States and some businesses have begun to fight the proliferation of their use.

Mr. Haag and other groups that train service animals for veterans have become advocates for changes to the law for taking animals on planes, and would like to see local or federal regulations that make animal certification a requirement for public spaces.

“There needs to be standards to hold people accountable,” said Lori Stevens, the founder of Patriot Paws, which works with Texas prisoners to train service dogs for veterans.