We present aperture masking interferometry (AMI) observations of the star HIP 65426 at 3.8 mu m, as part of the JWST Direct Imaging Early Release Science program, obtained using the Near Infrared Imager and Slitless Spectrograph instrument. This mode provides access to very small inner working angles (even separations slightly below the Michelson limit of 0.5 lambda/D for an interferometer), which are inaccessible with the classical inner working angles of the JWST coronagraphs. When combined with JWST's unprecedented infrared sensitivity, this mode has the potential to probe a new portion of parameter space across a wide array of astronomical observations. Using this mode, we are able to achieve a 5 sigma contrast of Delta mF380M similar to 7.62 +/- 0.13 mag relative to the host star at separations greater than or similar to 0 .'' 07, and the contrast deteriorates steeply at separations less than or similar to 0 .'' 07. However, we detect no additional companions interior to the known companion HIP 65426b (at separation similar to 0 .'' 82 or 87-31+108au ). Our observations thus rule out companions more massive than 10-12 MJup at separations similar to 10-20 au from HIP 65426, a region out of reach of ground- or space-based coronagraphic imaging. These observations confirm that the AMI mode on JWST is sensitive to planetary mass companions at close-in separations (greater than or similar to 0 .'' 07), even for thousands of more distant stars at similar to 100 pc, in addition to the stars in the nearby young moving groups and associations, as stated in previous works. This result will allow the planning and successful execution of future observations to probe the inner regions of nearby stellar systems, opening an essentially unexplored parameter space.