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We present the results of an aperture-masking interferometry survey for substellar companions around 67 members of the young (similar to 8-200 Myr) nearby (similar to 5-86 pc) AB Doradus, beta Pictoris, Hercules-Lyra, TW Hya, and Tucana-Horologium stellar associations. Observations were made at near-infrared wavelengths between 1.2 and 3.8 mu m using the adaptive optics facilities of the Keck II, Very Large Telescope UT4, and Palomar Hale Telescopes. Typical contrast ratios of similar to 100-200 were achieved at angular separations between similar to 40 and 320 mas, with our survey being 100% complete for companions with masses below similar to 0.25 M(circle dot) across this range. We report the discovery of a 0.52 +/- 0.09 M(circle dot) companion to HIP 14807, as well as the detections and orbits of previously known stellar companions to HD 16760, HD 113449, and HD 160934. We show that the companion to HD 16760 is in a face-on orbit, resulting in an upward revision of its mass from M(2) sin i similar to 14 M(J) to M(2) = 0.28 +/- 0.04 M(circle dot). No substellar companions were detected around any of our sample members, despite our ability to detect companions with masses below 80 M(J) for 50 of our targets: of these, our sensitivity extended down to 40 M(J) around 30 targets, with a subset of 22 subject to the still more stringent limit of 20 M(J). A statistical analysis of our non-detection of substellar companions allows us to place constraints on their frequency around similar to 0.2-1.5 M(circle dot) stars. In particular, considering companion mass distributions that have been proposed in the literature, we obtain an upper limit estimate of similar to 9%-11% for the frequency of 20-80 M(J) companions between 3 and 30 AU at 95% confidence, assuming that their semimajor axes are distributed according to dN/da alpha a(-1) in this range.