Can a FAA "Letter of Authorization" be used when there are no pilots that are current on a type rating?
If I understand the question correctly;

Maybe

The FAA outlines a generic type rating curriculum in this AC which is worth going through as it outlines what training would potentially be required for the type cert, 

FAA FAR 61.31 does allow for a waiver 


  § 61.31 Type rating requirements, additional training, and authorization requirements.
  
  (b)Authorization in lieu of a type rating. A person may be authorized
  to operate without a type rating for up to 60 days an aircraft
  requiring a type rating, provided -

(1) The Administrator has authorized the flight or series of flights;

(2) The Administrator has determined that an equivalent level of
 safety can be achieved through the operating limitations on the
 authorization;

(3) The person shows that compliance with paragraph (a) of this
 section is impracticable for the flight or series of flights; and

(4) The flight -

      (i) Involves only a ferry flight, training flight, test flight, or
      practical test for a pilot certificate or rating;

      (ii) Is within the United States;

      (iii) Does not involve operations for compensation or hire unless the
       compensation or hire involves payment for the use of the aircraft for
       training or taking a practical test; and

      (iv) Involves only the carriage of flight crewmembers considered
         essential for the flight.

(5) If the flight or series of flights cannot be accomplished within
   the time limit of the authorization, the Administrator may authorize
   an additional period of up to 60 days to accomplish the flight or
   series of flights.



Section 3 is the important one. You would have to show that it is "impractical" for you to obtain the type rating with out the waiver to operate the aircraft and fly the required training time.

You may also have a compounded issue of then needing a second crew member who is typed in the aircraft and can provide training/co-pilot duties. Further more you cant conduct any commercial operations while doing this so you need an empty DC-4 and a lot of time (read money) to fly it. Keep in mind you may only have 60 days to accomplish all the training as well or you will need to get an extension on the waiver.  



Just a general note. These kinds of situations are often best suited for an aviation lawyer who has experience dealing with the FSDO. It would be wise in this case to seek the aid of someone like that. You also may want to look for remote training if there is another operational DC-4 flying elsewhere and an arrangement could be made for you to go train with them for some time that situation may, in the long run be simpler.