Well of course they aren't "” but that's constitutionally irrelevant: Corporations aren't "real people" in the sense that the Constitution's protection of sexual privacy or prohibition on slavery make no sense in this context, but that doesn't mean that corporate entities also lack, say, Fourth Amendment rights. Or would the "no rights for corporations" crowd be okay with the police storming their employers' offices and carting off their (employer-owned) computers for no particular reason? "” or to chill criticism of some government policy.
Or how about Fifth Amendment rights? Can the mayor of New York exercise eminent domain over Rockefeller Center by fiat and without compensation if he decides he'd like to move his office there?
So corporations have to have some constitutional rights or nobody would form them in the first place. The reason they have these rights isn't because they're "legal" persons, however "” though much of the doctrine builds on that technical point "” but instead because corporations are merely one of the ways in which rights-bearing individuals associate to better engage in a whole host of constitutionally protected activity.
That is, the Constitution protects these groups of rights-bearing individuals. The proposition that only human beings, standing alone, with no group affiliation whatsoever, are entitled to First Amendment protection "” that "real people" lose some of their rights when they join together in groups of two or ten or fifty or 100,000 "” is legally baseless and has no grounding in the Constitution. George Mason law professor Ilya Somin, also a Cato adjunct scholar, discusses this point here.