Corporations have always been fictive persons—or at least fictive bodies, given the root—in some sense or another. For profit corporations are fictive persons for financial purposes. If Corporation X owes me money and goes into default, I only have a right to some share of Corporation X's assets, as the fictive person involved, and not to the assets of all the shareholders of Corporation X. This is much like the idea that if you owe me money, I may be able to get it from your spouse in some states—because, in one legal sense, you really are one body, one corporate being with shared assets—but I cannot attempt to attach the assets of your parents or siblings or best friend, since they are distinct legal persons. So, the shareholders and even officers of a corporation are legally distinct from the fictive person of the corporation.
But, (five of) the Supreme Court Justices have told us that corporations and other fictive persons are persons with respect to the Bill of Rights. Inasmuch as they are such, it would seem to follow that they have the other rights (and responsibilities) of persons. It is important to note that the operative concept in both the common and the civil law is person and not human being, so we needn't worry about the fact that corporations are not humans.
So, quite apart from worrying about whether Coca-Cola has registered for the draft—wait, they are beyond that age now—we should ask our corporations to forego the middlemen of lobbyists and campaign contributions and merely run for office themselves. I offer for your consideration, the Senator from the State of California, Apple, the Mayor of New York, The Bloomberg Corporation (only a small step), and our new President, Berkshire Hathaway.