Despite the phrasing being "Congress shall make no law," the First Amendment absolutely applies to the executive.
The mainstream test just looks for a "state actor" but even if you want to go textualist, things probably break down like this: if a law empowers the executive to do an act that violates the First Amendment, the law itself is a violation of the First Amendment. If no law empowers the executive to do the action at issue, then the action is illegal by virtue of being in excess of the executive's authority.
Just to further clarify, by "mainstream" I meant "the actual test courts use." The "textualist" version is more something to write a law review article about, a theory of how you might get to roughly the same place we are in reality while giving weight to the word "Congress."
Take the state actor doctrine in combination with the fact that the First Amendment is incorporated against the states, and you get the reality that it even applies to, for example, state colleges and universities disciplining students and teachers, despite them not being "Congress," a "law," or even an agent of the federal government at all.
The mainstream test just looks for a "state actor" but even if you want to go textualist, things probably break down like this: if a law empowers the executive to do an act that violates the First Amendment, the law itself is a violation of the First Amendment. If no law empowers the executive to do the action at issue, then the action is illegal by virtue of being in excess of the executive's authority.