Since Regina Dugan became the director of Darpa, the Pentagon’s top research division has signed millions of dollars’ worth of contracts with her family firm, which in turn owes her at least a quarter-million dollars. It’s an arrangement that has raised eyebrows in the research community, and has now drawn the attention of the Defense Department’s internal auditors and investigators.

The Pentagon’s Inspector General is launching an audit of those deals — and of every other research contract Darpa has signed during Dugan’s two-year tenure. This is just “the first in a series of planned audits to...