No, Congress can't pass a law permitting a special election for president
UPDATE: Please read the update at the end of this piece! Several thoughtful responses have been sent to me, and I am reconsidering whether I have this right at all! Such is the half-baked musing of a blog... I deeply appreciate feedback.
I recently read a piece on the crisis of presidential succession. The logic of the piece went something like this: suppose Russia interfered so greatly in our presidential electoral process that Donald Trump, Mike Pence, Paul Ryan, Orrin Hatch, Rex Tillerson, and everyone else in the ordinary line of succession had their legitimacy cast into doubt, a "stolen" election? Congress ought to pass a law to address this point--in particular, if Congress removes the President and Vice President (N.B.: this is, of course, the same irretrievably-corrupted Congress in which Mr. Hatch is the President Pro Tempore and Mr. Ryan is the Speaker), it should be authorized to call (N.B.: again, Congress is controlled by the same corrupt Russian stooges) for a special presidential election.
Because others appear to be taking this argument seriously, it's worth noting that it should not. Congress lacks the power to call for a special election for president.
The purported ground for the exercise of this power arises from Article II, Section 1, Clause 6--at least, the portions not altered by the Twenty-Fifth Amendment. The relevant provision reads: ". . . and the Congress may by Law provide for the Case of Removal, Death, Resignation or Inability, both of the President and Vice President, declaring what Officer shall then act as President, and such Officer shall act accordingly, until the Disability be removed, or a President shall be elected."
The power of Congress is one to enact a law to "provide" for the case of the loss of both the President and the Vice President. But that "Law" may "provide" one thing: "declaring what Officer shall then act as President." That is the extent of Congress's power in this area, at least under this Clause.
The remainder of the Clause does not empower Congress to act further. "[S]uch officer shall act accordingly," the Clause explains, "until the Disability be removed, or a President shall be elected." Both are in the passive voice, and, I think, deliberately so. The last provision, "a President shall be elected," then would refer to the ordinary powers of Congress to "determine the Time of chusing the Electors, and the Day on which they shall give their votes." But as the term of office is for "four Years," Congress's power is, I think, limited to this--the "term" of the President ends after four years (now on January 20th, see Amendment XX, Section 1), and the Officer "act[s]" as president "until" the next election.
The attempt to read into these provisions a power of Congress to call a special election is still more deeply flawed. The Constitution speaks of the power to fill vacancies with the special term "writ of election." There is no such power of Congress to issue writs of election for vacancies in the office of President and Vice President.
Consider the language of the Constitution for the House. Article I, Section 2, Clause 1 provides, "The House of Representatives shall be composed of Members chosen every second Year by the People of the several States . . . ." And in Clause 4, "When vacancies happen in the Representation from any State, the Executive Authority thereof shall issue Writs of Election to fill such Vacancies." (Emphasis added.) The ordinary process differs from the power during a vacancy, which is to issue writs of election.
When legislatures elected Senators, there was a similar provision, albeit not for writs of election. Article I, Section 3, Clause 1 provides, "The Senate of the United States shall be composed of two Senators from each state, chosen by the Legislature thereof . . . ." And in the next clause, ". . . if Vacancies happen by Resignation, or otherwise, during the Recess of the Legislature of any State, the Executive thereof may make temporary Appointments until the next Meeting of the Legislature, which shall then fill such Vacancies." There is specific new power to fill vacancies if they arise.
The Seventeenth Amendment works the same way as the House's language. "The Senate of the United States shall be composed of two Senators from each state, elected by the people thereof, for six years . . . " In the second clause, "When vacancies happen in the representation of any State in the Senate, the executive authority of such State shall issue writs of election to fill such vacancies . . . " (Emphasis added.) Again, a specific enumeration of power in the event vacancies arise.
Congress has no such power to issue writs of election when vacancies arise in the office of President and Vice President. Indeed, Congress's power is carefully (and understandably) limited to choosing successors who can "act" as president until the next election. It has no power to otherwise fill vacancies.
Indeed, Congress's power to regulate presidential elections is even more limited than its power to regulate congressional elections. The Times, Places and Manner Clause permits Congress to "make or alter such Regulations" relating to the "Times, Places and Manner of holding Elections for Senators and Representatives." But its power to regulate the selection of presidential electors is to "determine the Time of chusing the Electors, and the Day on which they shall give their Votes." Even if one believes that the power to issue writs of election for vacancies is a "manner" of regulation an election ([Robert G. Natelson offers evidence of this][1]), Congress lacks such power in presidential elections. Indeed, such a lack of power in presidential elections is one of the great reasons Congress enacted the Twenty-Fourth Amendment to abolish poll taxes in federal elections--many in Congress who thought Congress could do so for congressional elections under the Times, Places and Manner Clause believed they lacked the power to do so in presidential primaries and the selection of electors. (This is part of ongoing research I hope to share soon.)
[1]: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1751720
So Congress can't just pass a law permitting it to call for a special election to redo the presidential election if it so desires. The power to issue a writ of election to fill the vacancies in the office of President and Vice President if they arose would need to occur by constitutional amendment.
(UPDATE: I've learned that there is much more history here than my analysis offered! Congress in 1792 enacted a law that would permit it to call for a special election in the event of a vacancy. There is apparently a rich body of debates surrounding this--suggesting I should do more research before blogging! Nevertheless, I feel fairly confident my "writ of election" point is a significant textual one, even if Congress has previously disputed it. More to come one day!)