"Customer" is correct.
Actually, I think "diner" is the most suitable word choice here, although "customer" is fine.
"Client" is wrong. In the United States, "client" is used only in a professional relationship, and only for certain professions. Lawyers have clients. Accountants have clients. Social workers have clients.
Oddly enough, though, the set of all of a business's customers can be called the business's "clientele."