We have used WFPC2 on board the Hubble Space Telescope to obtain images of 47 members of the Taurus and Chamaeleon I star-forming regions that have spectral types of M6-L0 (M~0.01-0.1 M_{sun}_). An additional late-type member of Taurus, FU Tau (M7.25+M9.25), was also observed with adaptive optics at Keck Observatory. In these images, we have identified promising candidate companions to 2MASS J04414489+2301513 ({rho}=0.105"/15 AU), 2MASS J04221332+1934392 ({rho}=0.05"/7 AU), and ISO 217 ({rho}=0.03"/5 AU). We reported the first candidate in a previous study, showing that it has a similar proper motion as the primary in images from WFPC2 and Gemini adaptive optics. We have collected an additional epoch of data with Gemini that further supports that result. By combining our survey with previous high-resolution imaging in Taurus, Chamaeleon I, and Upper Sco ({tau}~10 Myr), we measure binary fractions of 14/93=0.15_-0.03_^+0.05^ for M4-M6 (M~0.1-0.3 M_{sun}_) and 4/108=0.04_-0.01_^+0.03^ for >M6 (M<~0.1 M_{sun}_) at separations of >10 AU. Given the youth and low density of these regions, the lower binary fraction at later types is probably primordial rather than due to dynamical interactions among association members. The widest low-mass binaries (>100 AU) also appear to be more common in Taurus and Chamaeleon I than in the field, which suggests that the widest low-mass binaries are disrupted by dynamical interactions at >10 Myr, or that field brown dwarfs have been born predominantly in denser clusters where wide systems are disrupted or inhibited from forming.