We can see “who…” modifier following the underlined portion. Logically people should precede “who”, not an organization. Keeping this in mind let’s check the answer choices.
A. by NASA's predecessor, the National Advisory Committee for Aeronautics, as so-called human computers,
If an inessential appositive in between “the National… Aeronautics” is removed, we have “hired as human computers, who…” - correct modification.
B. to serve as so-called human computers by the National Advisory Committee for Aeronautics, NASA's predecessor,
“who” illogically tries to modify either “NASA's predecessor” or “the National Advisory Committee for Aeronautics”, both of which are ineligible referents.
C. to perform the function of so-called human computers by NASA's predecessor, the National Advisory Committee for Aeronautics
“who” illogically tries to modify either “NASA's predecessor” or “the National Advisory Committee for Aeronautics”, both of which are ineligible referents.
D. so that they can serve as so-called human computers by NASA's predecessor, the National Advisory Committee for Aeronautics,
“who” illogically tries to modify either “NASA's predecessor” or “the National Advisory Committee for Aeronautics”, both of which are ineligible referents.
E. by the predecessor of NASA, which was called the National Advisory Committee for Aeronautics, as so-called human computers,
“which” may refer to either “predecessor” or “NASA” – ambiguous.
So A