Millimeter observations of disks around young stars reveal substructures indicative of gas pressure traps that may aid grain growth and planet formation. We present Submillimeter Array observations of HD 34700- two Herbig Ae stars in a close binary system (Aa/Ab, $sim$0.25 AU), surrounded by a disk presenting a large cavity and spiral arms seen in scattered light, and two distant, lower mass companions. These observations include 1.3 mm continuum emission and the $^{12}$CO 2-1 line at $sim0.5$ (178 AU) resolution. They resolve a prominent azimuthal asymmetry in the continuum, and Keplerian rotation of a circumbinary disk in the $^{12}$CO line. The asymmetry is located at a radius of $155^{+11}_{-7}$ AU, consistent with the edge of the scattered light cavity, being resolved in both radius ($72 ^{+14}_{-15}$ AU) and azimuth (FWHM = $64 ^{circ +8}_{-7}$). The strong asymmetry in millimeter continuum emission could be evidence for a dust trap, together with the more symmetric morphology of $^{12}$CO emission and small grains. We hypothesize an unseen circumbinary companion, responsible for the cavity in scattered light and creating a vortex at the cavity edge that manifests in dust trapping. The disk mass has limitations imposed by the detection of $^{12}$CO and non-detection of $^{13}$CO. We discuss its consequences for the potential past gravitational instability of this system, likely accounting for the rapid formation of a circumbinary companion. We also report the discovery of resolved continuum emission associated with HD 34700B (projected separation $sim1850$AU), which we explain through a circumstellar disk.
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