Former federal prosecutor and frequent plain-language law explainer Ken "Popehat" White has done the (American) Internet the immense service of producing a master(ful) post about the First Amendment, explaining why the American constitutional basis for free speech includes abridgments on speech by some private actors and why it can be invoked in civil cases.
If you said something like "The First Amendment says 'Congress shall make no law,' and Congress isn't involved here, so it's not a First Amendment issue."
Congratulations! You've read the First Amendment. Even if you've ignored the last century of discourse about it, this raises you above most of the populace, particularly on the Internet.
You're right that the plain language of the First Amendment only limits Congressional power. But you've ignored some American history. Don't worry: you've only ignored a century and a half of it. The Bill of Rights was originally understood to limit the power of the federal government without limiting the states. But in 1868, after some recent unpleasantness, we amended the Constitution to add the Fourteenth Amendment, which includes this language: "No state shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any state deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws." By its own terms, the Fourteenth Amendment forbids the states from infringing certain rights.
But which rights? Well, in the early 20th Century, the United States Supreme Court decided that certain fundamental rights enumerated in the Bill of Rights are included in the concept of liberty identified in the Fourteenth Amendment and therefore protected from infringement by "due process of law." This process — under which the court decided that the Fourteenth Amendment incorporated by reference rights from the Bill of Rights and made them enforceable against the states — is called incorporation, and the notion is called the incorporation doctrine. The Supreme Court has decided that most, but not all, rights from the Bill of Rights are incorporated by the Fourteenth Amendment and therefore protected from infringement by states. The Supreme Court decided — or, to be more accurate, assumed — in a 1925 decision that freedom of speech under the First Amendment is one of the rights enumerated in the Bill of Rights that the states may not infringe because it is incorporated by the due process clause of the Fourteenth Amendment.
Therefore, the First Amendment does apply to actions by states and their political subdivisions (cities, counties, state agencies, etc.), and has for more than ninety years. You're wrong. Please stop being wrong and asking other people to be wrong with you.
Hello! You've Been Referred Here Because You're Wrong About The First Amendment. [Ken White/Popehat]
(Image: First amendment area Muir Woods, Brandt Luke Zorn, CC-BY-SA)