There is no "right to privacy" per se in the U.S. There is a right to be free of unreasonable searches and seizures (4th amendment), there is a right to refuse to testify against yourself at trial (5th amendment), etc. These protect certain aspects of a right to privacy, but if you read writings from the founding generation, they didn't have the kind of well-developed "right to privacy" concept many people have today.

As a result, these protections exist next to mechanisms that cut in the other direction. One of those mechanisms is the almost unlimited power of common law courts to compel people to furnish information, through subpoenas and warrants. And that power is especially strong when the information is about someone else. So for example, while a court cannot force you to testify against yourself (5th amendment), they can force you to testify against someone else. They can force you to furnish information or copies of documents relevant to an investigation or ongoing litigation. Indeed, in a civil litigation, often one of the lawyers will be able to issue subpoenas forcing parties to turn over documents in the name of the court. In the Anglo-American system, there is really no "right of privacy" as against the right of a court to gather all the evidence necessary in a civil or criminal litigation.

So corporations do have 4th amendment rights. The NSA doesn't come to their offices and take the information. But that doesn't protect them against court orders to furnish information about other people.

Citizens United did not say that corporations are people. If you are starting from that misunderstanding then lots of things won't make sense. Corporate personhood is not mentioned in the decision.

A year after Citizens the court ruled unanimously in FCC v ATT that corporations fail to meet the "personal" part of "personal privacy." That decision is fun to read if only for the last paragraph.