Emotional Support Animals don't have the same rules and regulations applied to them as other assistance Animals such as dogs that assist people with disabilities and don't have the same training so a public place can bar the emotional support animal from entering if it so desires.
There's no law that can force any public place to allow an emotional support animal to accompany the owner.
An exception though as long as your emotional support animal is registered as an emotional support animal and you can provide proof then an airline must allow the emotional support animal to fly along with you.