The First Amendment protects the right to organize in addition to free speech. The NLRB (and the Wagner Act, the law that gives us the right to discuss wages, as well as unionize, without retaliation) have a storied history of being challenged on first amendment grounds.
People have tried arguing that an employer’s first amendment rights are violated by a law that prevents them from firing someone for any reason they want. The government codifying what an employer can and can’t fire an employee for is directly related to the first amendment.
Any time you’re talking about protected speech, or the right to organizing, its directly related to the first amendment. If you can’t see that, then I don’t know what to tell you.