Could an acting president appoint a vice president?

[Slander and/or Libel]

There are some concerns about the constitutionality of the 1947 Presidential Succession Act as well, and whether legislative officers constitutionally qualify as constitutional Officers capable of acting as President.

To wit:

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If the "officers" mentioned in Article II, Section 1, Clause 6 are the same as the "officers of the United States" mentioned in Section 2, Clause 2 and Section 3, then it's pretty apparent that legislative officers don't belong on that list, and that having the Speaker and President Pro Tempore in the line of succession is unconstitutional. This is all a completely unresolved question, though, obviously.

Also, and I know this is somewhat off topic, the 1947 Presidential Succession Act is a total mess. The Speaker and President Pro Tempore have to resign to act as president. It's unclear whether they have the ability to willingly pass on being Acting President if called upon, or how long they'd have to make their choice. If they don't have the ability to pass, then for instance, Obama and Biden could have scheduled colonoscopies for the same day and forced Paul Ryan and Orrin Hatch from office. Of course, if they did so, Kerry would have been forced to act as president, and in so doing resign from his post as Secretary of State. How does it make sense for a cabinet secretary, who is in line to act as president by virtue of holding a cabinet post, to have to resign that post to act as president? When the Vice President acts as president he's not required to resign the Vice Presidency. Why should a Cabinet official? Note: previous versions of the Presidential Succession Act did not do this. Previous versions also specified a means for a special election in the case of a double vacancy.

[Slander and/or Libel]

Probably because the Vice President, when acting as president, is only temporarily in charge. He only acts as president when the latter can't execute his powers, but is de facto still the president. For example during a surgery. If the vice president succeeds to the president, is also required to "leave" the vice presidential office (I don't know whether a vice president could refuse to take over). The difference is that he assumes the role for the rest of term. So if a Speaker had to be acting president, he'd likely stay in power until the next regular election.

But the Succession Act also provides that a cabinet official can be bumped by a later qualifying Speaker or President Pro Tempore. It's clear why the Speaker or PPT would have to resign, and that's because they couldn't simultaneously be members of the legislative and executive branches, constitutionally. That's clear. But it's unclear why a cabinet official who, by virtue of being a cabinet official, assumes the acting presidency, should have to resign the cabinet position to do so, especially if there is a mechanism for chicanery on the part of the legislature explicitly written into the law.