Final answer:
The Joint Chiefs of Staff is the agency that provides deployed commanders with general information or specific inquiries across the entire spectrum of personnel programs.
Step-by-step explanation:
The agency that can provide deployed commanders and their troops with information or respond to specific inquiries across the entire spectrum of personnel programs is most likely the Joint Chiefs of Staff (JCS). The JCS consists of high-ranking military officers from the Army, Navy, Air Force, and Marine Corps, who work alongside the chair and vice-chair to advise the President on military matters and coordinate military efforts. This body is well-positioned to offer guidance and support to deployed commanders, given that it oversees the many elements of the U.S. armed forces and formulates policies concerning the United States' military affairs.