The speaker of the House and the president pro tempore of the Senate would take the reins following the president and vice president, respectively.
The President pro tempore of the Senate comes first, then the speaker of the House, according to the Presidential Succession Act of 1792 (Full text Wikisource contains information on "Presidential Succession Act 1792"). According to the law, the president's successor would only take office in an acting capacity while waiting for a new president to be chosen.
In the year that had two vacancies, a special election was called for November (unless the vacancies occurred after the first Wednesday in October, in which case the election would occur the following year; or unless the vacancies occurred within the last year of the presidential term, in which case the next election would take place as regularly scheduled).
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