There are some exceptions to the general principle (discussed in my last post) that there be some state action involved in order for the federal or a state constitutional provision to be violated. First, there are some early cases that suggest that private actors who are performing government-type functions could be constitutionally barred (e.g., 19th century caselaw on "company towns"). Second, some state constitutions have been read to bar certain private actors from restricting free speech rights of the public (e.g., the mall cases in NJ). Finally, some recent cases have held that homeowners' associations and similar entities are barred by the Constitution from restricting the political speech of its members even where the members have signed contracts agreeing to such restrictions (e.g., no political campaign posters on front lawns).
None of these limited exceptions to the genral rule would appear to apply to chess.com.
The First Amendment prohibits Congress from passing a law abridging the freedom of speech. It doesn't prohibit Chess.com from censoring material that you submit to their site.
Well, by virtue of the Fifth Amendment, the First Amendment also prohibits states from abridging freedom of speech. And caselaw clearly indicates that executive and/or judicial action abridging free speech is prohibited by the Constitution. But your main point is correct: some form of state action is required to find a constitutional violation. Private actors generally can do what they want (e.g., censorship) without violating the First Amendment.