Up until 2012, Article 149 of the Haitian Constitution used to obligate the National Assembly, convened by the Prime Minister, to invest the President of the Supreme Court of the Republic (or, if unavailable, another senior judge in order of seniority) with the duties of President of the Republic - with new elections to be held between 45 and 90 days later.
However, the constitution was amended in 2012, and now reads:
In case of vacancy of the Presidency of the Republic either by
resignation, dismissal, death or in case of physical or mental
permanent incapacity duly declared, the Council of Ministers,
under the presidency of the Prime Minister, exercises
the Executive Power until the election of another President.
case, the ballot for the election of the new President of
the Republic for the time that remains to complete the mandate
takes place sixty (60) days at least and one hundred twenty
(120) days at most after the beginning of the vacancy,
in accordance with the Constitution and the electoral law.
In the case
where the vacancy is produced from the fourth year of the
presidential mandate, the National Assembly meets of office
within the sixty (60) days which follow the vacancy to elect
a new Provisional President of the Republic for the time that
remains to complete the mandate.
So, in this case, the Council of Ministers, led by interim Prime Minister Claude Joseph exercises executive power until the office of president is filled.
As President Moïse began his mandate on February 7th, 2017, the last paragraph applies, and the National Assembly (a joint session of the two houses of parliament) will within sixty days elect a new provisional president until the end of Moïse's mandate in February 2022.
Two issues complicate the matter - firstly, President Moïse had named Claude Joseph's replacement, Ariel Henry, on July 5th. Although Henry had not been sworn in, the Constitution doesn't mention any such requirement to take office. Henry's supporters argue that this makes him the legitimate Prime Minister.
Secondly, Moïse was elected in the 2015 elections. He would normally have taken office on February 7th, 2016, however, the run-off election was delayed until October 2016 - with an interim President holding office in the meantime. Opponents of Moïse suggest that his mandate should have expired on February 7th, 2021, five years after he should have taken office. The relevant article in the constitution is Article 134-2:
The presidential election takes place on the last Sunday of
October of the fifth year of the presidential mandate.
president elected enters into his functions on 7 February
following the date of his election. In the case where the
ballot cannot take place before 7 February, the president
elected enters into his functions immediately after the
validation of the ballot and his mandate is considered to have
commenced on 7 February of the year of the election.