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The Vela and Puppis A supernova remnants (SNRs) comprise a large emission region of $\sim 8^{\circ}$ diameter in the soft X-ray sky. The HaloSat CubeSat mission provides the first soft X-ray ($0.4-7$ keV) observation of the entire Vela SNR and Puppis A SNR region with a single pointing and moderate spectral resolution. HaloSat observations of the Vela SNR are best fit with a two-temperature thermal plasma model consisting of a cooler component with $kT_{1} = 0.19^{+0.01}_{-0.01}$ keV in collisional ionization equilibrium and a hotter component with $kT_{2} = 1.06^{+0.45}_{-0.27}$ keV in non-equilibrium ionization. Observations of the Puppis A SNR are best fit with a single-component plane-parallel shocked plasma model with $kT = 0.86^{+0.06}_{-0.05}$ keV in non-equilibrium ionization. For the first time, we find the total X-ray luminosities of both components of the Vela SNR spectrum in the $0.5-7$ keV energy band to be $L_X = 4.4^{+1.4}_{-1.4} \times 10^{34}$ erg s$^{-1}$ for the cooler component and $L_X = 4.1^{+1.8}_{-1.5} \times 10^{34}$ erg s$^{-1}$ for the hotter component. We find the total X-ray luminosities of the Vela and Puppis A SNRs to be $L_{\text{X}} = 8.4 \times 10^{34}$ erg s$^{-1}$ and $L_X = 6.7^{+1.1}_{-0.9} \times 10^{36}$ erg s$^{-1}$.