(1) The Vacancies Act is unconstitutional, and no acting appointment can be made, so whoever is next in the the line of succession will run the agency until a full appointment is made and confirmed (Rosenstein, and nobody else, unless Rosenstein is fired by the President).
(2) The Vacancies Act is constitutional, and the President can appoint anybody confirmed for anything by the Senate(e.g., an ambassador) or anybody who has worked at Justice for over 90 days at more than GS-15 pay, or the AG’s first assistant.
Note that if (1) is true, then it is also true for new Administrations, and the new President must let the old political appointees run the agency or fire them all, in which case 28 U.S. Code § 508 – Vacancies runs out of applicability and presumably, without any statute, succession runs to the senior civil servant or some other civil servant in the agency whom the President appoints, or the President runs the agency directly.