We present a search for new planetary-mass members of nearby young moving groups (YMGs) using astrometry for 694 T and Y dwarfs, including 447 objects with parallaxes, mostly produced by recent large parallax programs from UKIRT and Spitzer. Using the BANYAN Sigma and LACEwING algorithms, we identify 30 new candidate YMG members, with spectral types of T0-T9 and distances of 10-43 pc. Some candidates have unusually red colors and/or faint absolute magnitudes compared to field dwarfs with similar spectral types, providing supporting evidence for their youth, including four early-T dwarfs. We establish one of these, the variable T1.5 dwarf 2MASS J21392676+0220226, as a new planetary-mass member (14.6(-1.6)(+3.2) M-Jup) of the Carina-Near group (200 +/- 50 Myr) based on its full six-dimensional kinematics, including a new parallax measurement from CFHT. The high-amplitude variability of this object is suggestive of a young age, given the coexistence of variability and youth seen in previously known YMG T dwarfs. Our four latest-type (T8-T9) YMG candidates, WISE J031624.35+430709.1, ULAS J130217.21+130851.2, WISEPC J225540.74-311841.8, and WISE J233226.49-432510.6, if confirmed, will be the first free-floating planets (approximate to 2-6 M-Jup) whose ages and luminosities are compatible with both hot-start and cold-start evolutionary models, and thus overlap with the properties of the directly imaged planet 51 Eri b. Several of our early/mid-T candidates have peculiar near-infrared spectra, indicative of heterogenous photospheres or unresolved binarity. Radial velocity measurements needed for final membership assessment for most of our candidates await upcoming 20-30 m class telescopes. In addition, we compile all 15 known T7-Y1 benchmarks and derive a homogeneous set of their effective temperatures, surface gravities, radii, and masses.