No. The First Amendment restricts governmental action only.
We are contacted frequently by people complaining that Facebook, Twitter, Intagram, or other private corporate social-media companies supposedly violated their "free-speech" or "First Amendment" rights. These potential clients want us to "sue" these social-media companies for them.
But Facebook, Twitter, and Instagram are private companies. They are not the government. They can set their own rules on speech and there's nothing you can do about it. A court that tried to do something about it would be violating those companies' right to free speech. And conservative judges have ruled—and conservative politicians have insisted—that private corporations have a right to free speech.
Others contact us complaining that some private store violated their "free-speech" rights by requiring face masks during the pandemic, or barring political speech of one kind or another, or barring political speech generally.
There's nothing we can do about that legally, even if a client paid hourly rates out of pocket. We couldn't even file a complaint on that issue in good faith, the issue is so crystal-clear in the law. For some reason, there are broad misconceptions in the public about this—an assumption that we are free to say anything anywhere, anytime, in any forum, even private ones. That's just not true.
If clients can show is clear evidence that private, non-governmental actors are acting in concert or conspiracy with government actors to restrict speech, we might be able to bring a civil-rights-conspiracy claim. But that would be a rare exception to the rule above. The evidence would have to be strong.