I was researching Harry Truman and came across this window between 1945 and 1947 (the passage of the act) where there seems to be no protocol for what to do upon a president's death if he has no vice president.
I hope this question doesn't fall into the category of alternate history, but if it does, let me know and I'll take it down.
There was a law at the time, and had been starting from the Washington administration: the two previous Presidential Succession Acts. From 1886 to 1947, the order of succession was the Cabinet, in order of creation of the departments (just like the 1947 act), but stopping at Interior. The 1886 act removed the requirement of a special Presidential election.
It took me about five minutes to find sources -- Edit: I mention this just to show my work and in case someone wants to know about a simple lookup:
I did a Google-brand Web search for "Presidential Succession Act". One of the top hits was the Wikipedia article Presidential Succession Act.
Edit: Please don't hurt me -- please note that I'm using it as a possible lead for other things to look for and for something that might be an overview.
The law at the time was the Presidential Succession Act of 1886, which had modified the previous Presidential Succession Act of 1792.
Searching for "Presidential Succession Act of 1886" in Google had, as its third hit, a Wikisource article Presidential Succession Act 1886. It links to the 1792 acts, as it repealed certain provisions, but it's hard to tell which from those texts: the 1886 source refers to sections of the Revised Statutes, but the 1792 source isn't numbered that way. You can find the applicable sections here: the 1886 act repealed sections 9 and on of the 1792 act, except section 11, the requirement that a resignation can be done only by explicit written notification to the Secretary of State. Edit: my search was "revised statutes" "united states"