We present the first results from a recently concluded study of GRBs at z>~5 with the Karl G. Jansky Very Large Array (VLA). Spanning 1 to 85.5GHz and 7 epochs from 1.5 to 82.3d, our observations of GRB140311A are the most detailed joint radio and millimeter observations of a GRB afterglow at z>~5 to date. In conjunction with optical/near-IR and X-ray data, the observations can be understood in the framework of radiation from a single blast wave shock with energy E_K,iso_~8.5x10^53^erg expanding into a constant density environment with density, n_0_~8cm^-3^. The X-ray and radio observations require a jet break at t_jet_~0.6d, yielding an opening angle of {theta}jet~4{deg} and a beaming-corrected blast wave kinetic energy of E_K_~2.2x10^50^erg. The results from our radio follow-up and multiwavelength modeling lend credence to the hypothesis that detected high-redshift GRBs may be more tightly beamed than events at lower redshift. We do not find compelling evidence for reverse shock emission, which may be related to fast cooling driven by the moderately high circumburst density.