We present the results of an aperture-masking interferometry survey for substellar companions around 67 members of the young (∼8-200Myr) nearby (∼5-86pc) AB Doradus, β Pictoris, Hercules-Lyra, TW Hya, and Tucana-Horologium stellar associations. Observations were made at near-infrared wavelengths between 1.2 and 3.8μm using the adaptive optics facilities of the Keck II, Very Large Telescope UT4, and Palomar Hale Telescopes. Typical contrast ratios of ∼100-200 were achieved at angular separations between 40 and 320mas, with our survey being 100% complete for companions with masses below 0.25M ⊙ across this range. We report the discovery of a 0.52 ± 0.09M ⊙ companion to HIP14807, as well as the detections and orbits of previously known stellar companions to HD16760, HD113449, and HD160934. We show that the companion to HD16760 is in a face-on orbit, resulting in an upward revision of its mass from M 2sin i 14M J to M 2 = 0.28 ± 0.04M ⊙. No substellar companions were detected around any of our sample members, despite our ability to detect companions with masses below 80M J for 50 of our targets: of these, our sensitivity extended down to 40M J around 30 targets, with a subset of 22 subject to the still more stringent limit of 20M J. A statistical analysis of our non-detection of substellar companions allows us to place constraints on their frequency around ∼0.2-1.5M ⊙ stars. In particular, considering companion mass distributions that have been proposed in the literature, we obtain an upper limit estimate of ∼9%-11% for the frequency of 20-80M J companions between 3 and 30AU at 95% confidence, assuming that their semimajor axes are distributed according to dN/da ∞ a-1 in this range.