No, we cannot do that, under current law. There are three things that regulate the line of succession: the 25th Amendment to the US Constitution, Article II Section 1, Clause 6 of the Constitution, and the Presidential Succession Act of 1947.
The relevant text of each:
25th Amendment (Section 1): In case of the removal of the President from office or of his death or resignation, the Vice President shall become President.
Article II, Section 1, Clause 6: In Case of the Removal of the President from Office, or of his Death, Resignation, or Inability to discharge the Powers and Duties of the said Office, the Same shall devolve on the Vice President, 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.
Note that the 25th Amendment clarifies the first part of II, 1, 6 by replacing “…the Same shall devolve on the Vice President…” with “…shall become President.” Traditionally, that’s how it’s always happened, but the first time a president died in office, in 1841, there were two days of debate over whether or not the VP had actually become president, or was merely acting as president, before Congress said “He’s the prez.” (The part of the original text about disability of a president is covered in other parts of the 25th, but that’s not relevant to your question).
The part of Clause 6 that says “…Congress may by Law provide for the case of…Death…both of the President and Vice President…” was done with the passage of the Presidential Succession Act of 1947.
The pertinent parts of that Act read (and I am editing for brevity and clarity): (1) If, by reason of death, resignation, removal from office, inability, or failure to qualify, there is neither a President nor Vice President to discharge the powers and duties of the office of President, then the Speaker of the House of Representatives shall, upon his resignation as Speaker and as Representative in Congress, act as President.
(2) The same rule shall apply in the case of the death, resignation, removal from office, or inability of an individual acting as President under this subsection.
(b) If, at the time when under subsection (a) of this section a Speaker is to begin the discharge of the powers and duties of the office of President, there is no Speaker, or the Speaker fails to qualify as Acting President, then the President pro tempore of the Senate shall, upon his resignation as President pro tempore and as Senator, act as President.
(c) An individual acting as President under subsection (a) or subsection (b) of this section shall continue to act until the expiration of the then current Presidential term ….
(1) If, by reason of death, resignation, removal from office, inability, or failure to qualify, there is no President pro tempore to act as President under subsection (b) of this section, then the officer of the United States who is highest on the following list, and who is not under disability to discharge the powers and duties of the office of President shall act as President: Secretary of State, Secretary of the Treasury, Secretary of Defense, Attorney General, Secretary of the Interior, Secretary of Agriculture, Secretary of Commerce, Secretary of Labor, Secretary of Health and Human Services, Secretary of Housing and Urban Development, Secretary of Transportation, Secretary of Energy, Secretary of Education, Secretary of Veterans Affairs, Secretary of Homeland Security.
(2) An individual acting as President under this subsection shall continue so to do until the expiration of the then current Presidential term….
So, the law that covers what happens if there’s neither a POTUS or a VP specifically says that whoever ends up taking over acts as President til the end of the then-current term. There are some scenarios where someone could end up bumping someone else out, but there is no provision for calling a special election under any circumstances.
Interestingly, though, prior to the 1947 Succession Act, the previous ones, which were passed in 1792 and 1886, DID state that, if such a vacancy happened prior to the start of the last year of a presidential term, there WOULD be a special election for a new POTUS and VP, who would then start a new, four-year term after being elected. That provision was obviously never enacted, and the current law from 1947 specifically and clearly states there will NOT be a special election.
Interestingly, when this Act could potentially have been enacted in 1973 (after the resignation of Spiro Agnew as VP and the appointment of Gerald Ford to replace him), and in 1974 (when Ford became POTUS after Nixon’s resignation, and the appointment of Nelson Rockefeller as VP), the then-Speaker of the House, Carl Albert, was a Democrat (and Nixon, Ford, and Rockefeller were Republicans)—during the times while the VP was vacant, and Speaker Albert was next in line, he publicly stated that, if he were to become Acting President, he would appoint a Republican Vice President and then resign, allowing that VP to become POTUS. Albert argued that the people had voted for a Republican president, and he didn’t want to have the presidency switch parties mid-term.
The other two times where there was a VP vacancy since 1947—from the date the Act went into effect through January 20, 1949 (VP vacant due to the elevation of Harry Truman to the presidency when FDR died) and from November 22, 1963-January 20, 1965 (VP vacant due to the elevation of LBJ after JFK was assassinated), the Speaker was from the opposite party as the President in the first situation, and the same as the President in the second. (Note that the reason no new VP was appointed in those situations is because the procedure for appointing a replacement VP was also contained in the 25th Amendment, which was ratified in 1967—prior to that, it was left vacant until the next election).