I construe a right of action to be the ability to sue somebody for injunctive and monetary relief. If there isn't a right of action for the First Amendment like how Bivens provided a right of action for the 4th Amendment, how is anybody able to sue a federal official for violating their first amendment rights?
In Egbert v. Boule, 596 U.S. ___ (2022), the Supreme Court held that "there is no Bivens action for First Amendment retaliation." The Court was unanimous on this point.
This means that right now, there is no way to sue for a damages remedy for First Amendment retaliation by federal officials. The majority said that "Congress, not the courts, is better suited to authorize such a damages remedy."