The Askaryan Radio Array (ARA) is an ultra-high energy (UHE) neutrino telescope at the South Pole consisting of an array of radio antennas with a bandwidth of 150-850 MHz buried under the Antarctic ice at ~200m depth aimed at detecting the Askaryan radiation produced by neutrino interactions in the ice. Currently, the experiment has five stations in operation that have been deployed in stages since 2012. This contribution focuses on the development of a search for a diffuse flux of neutrinos in two ARA stations (A2 and A3) from 2013-2016. A background of ~0.01-0.02 events is expected in one station in each of two search channels in horizontal- and vertical-polarizations. The expected new constraints on the flux of ultra-high energy neutrinos based on four years of analysis with two stations improve on the previous limits set by ARA by a factor of about two. The projected sensitivity of ARA's five-station dataset is beginning to be competitive with other neutrino telescopes at high energies near $10^{10.5}$ GeV.