National Air and Space Museum, Smithsonian Institution www.airandspace.si.edu [Air, Space]

Brief history http://airandspace.si.edu/about/history/.

Object collections, with online search instructions http://airandspace.si.edu/collections/archival/.

Archival collections with online search instructions http://airandspace.si.edu/collections/archival/.

Administrative records, including exhibits, are kept by the Smithsonian Archives http://siarchives.si.edu/.

Research facilities are described in http://airandspace.si.edu/research/,
including fellowship opportunities specific to the museum.

In addition the Smithsonian fellowship program covers all disciplines of interest to the Institution, including all aspects of history of science and technology. An overview is available in Smithsonian Opportunities for Research and Study, http://www.smithsonianofi.com/sors-introduction/, which provides a link to the PDF document; NASM is on pages 32-40 of the current (2018) edition and includes a staff diretory with contact information.
For details on how to apply see http://www.smithsonianofi.com/fellowship-opportunities/smithsonian-institution-fellowship-program/