The Wide Field Infrared Survey Telescope (WFIRST) Coronagraph Instrument (CGI) will be the first high-performance stellar coronagraph using active wavefront control for deep starlight suppression in space, providing unprecedented levels of contrast and spatial resolution for astronomical observations in the optical. One science case enabled by the CGI will be taking visible images and (R∼50) spectra of faint interplanetary dust structures present in the habitable zone of nearby sunlike stars (∼10 pc) and within the snow-line of more distant ones (∼20 pc), down to dust brightness levels commensurate with that of the solar system zodiacal cloud. Reaching contrast levels below 10-7 at sub-arcsecond angular scales for the first time, CGI will cross an important threshold in debris disks physics, accessing disks with low enough optical depths that their structure is dominated by transport mechanisms rather than collisions. Hence, CGI will help us understand how exozodiacal dust grains are produced and transported in low-density disks around mature stars. Additionally, CGI will be able to measure the brightness level and constrain the degree of asymmetry of exozodiacal clouds around individual nearby sunlike stars in the optical, at the ∼3x solar zodiacal emission level. This information will be extremely valuable for optimizing the observational strategy of possible future exo-Earth direct imaging missions, especially those planning to operate at optical wavelengths as well, such as the Habitable Exoplanet Observatory (HabEx) and the Large Ultraviolet/Optical/Infrared Surveyor (LUVOIR).