We present a search for prompt radio emission associated with the short-duration gamma-ray burst (GRB) 150424A using the Murchison Widefield Array (MWA) at frequencies from 80-133 MHz. Our observations span delays of 23 s-30 min after the GRB, corresponding to dispersion measures of 100-7700 pc/cm^3. We see no excess flux in images with timescales of 4 s, 2 min, or 30 min, and set a 3 sigma flux density limit of 3.0 Jy at 132 MHz on the shortest timescales: some of the most stringent limits to date on prompt radio emission from any type of GRB. We use these limits to constrain a number of proposed models for coherent emission from short-duration GRBs, although we show that our limits are not particularly constraining for fast radio bursts because of reduced sensitivity for this pointing. Finally, we discuss the prospects for using the MWA to search for prompt radio emission from gravitational wave transients and find that while the flux density and luminosity limits are likely to be very constraining, the latency of the gravitational wave alert may limit the robustness of any conclusions.
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