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We present a low-frequency (170\textendash200\,MHz) search for prompt radio emission associated with the long GRB 210419A using the rapid-response mode of the Murchison Widefield Array (MWA), triggering observations with the Voltage Capture System (VCS) for the first time. The MWA began observing GRB 210419A within 89\,s of its detection by \textit{Swift}, enabling us to capture any dispersion delayed signal emitted by this GRB for a typical range of redshifts. We conducted a standard single pulse search with a temporal and spectral resolution of $100\,\upmu$s and 10\,kHz over a broad range of dispersion measures from 1 to $5000\,\text{pc}\,\text{cm}^{-3}$, but none were detected. However, fluence upper limits of $77\text{--}224$\,Jy\,ms derived over a pulse width of $0.5\text{--}10$\,ms and a redshift of $0.6<z<4$ are some of the most stringent at low radio frequencies. We compared these fluence limits to the GRB jet-interstellar medium (ISM) interaction model, placing constraints on the fraction of magnetic energy ($ε_{\text{B}}\lesssim[0.05 \text{--} 0.1]$). We also searched for signals during the X-ray flaring activity of GRB 210419A on minute timescales in the image domain and found no emission, resulting in an intensity upper limit of $0.57\,\text{Jy}\,\text{beam}^{-1}$, corresponding to a constraint of $ε_{\text{B}}\lesssim10^{-3}$. Our non-detection could imply that GRB 210419A was at a high redshift, there was not enough magnetic energy for low-frequency emission, or that the radio waves did not escape from the GRB environment.