High-precision measurements of superallowed ${0}^{+}\ensuremath{\rightarrow}{0}^{+}$ $\ensuremath{\beta}$ decays of $T=2$ nuclides such as $^{20}\mathrm{Mg}$, $^{24}\mathrm{Si}$, $^{28}\mathrm{S}$, and $^{32}\mathrm{Ar}$ can contribute to searches for physics beyond the standard model of particle physics if the ${Q}_{\mathrm{EC}}$ values are accurate to a few keV or better. As a step toward providing precise ${Q}_{\mathrm{EC}}$ values for these decays, the ground-state masses of the respective daughter nuclei $^{20}\mathrm{Na}$, $^{24}\mathrm{Al}$, $^{28}\mathrm{P}$, and $^{32}\mathrm{Cl}$ have been determined by measuring the ($^{3}\mathrm{He},t$) reactions leading to them with the $^{36}\mathrm{Ar}$($^{3}\mathrm{He},t$)$^{36}\mathrm{K}$ reaction as a calibration. A quadrupole-dipole-dipole-dipole (Q3D) magnetic spectrograph was used together with thin ion-implanted carbon-foil targets of $^{20}\mathrm{Ne}$, $^{24}\mathrm{Mg}$, $^{28}\mathrm{Si}$, $^{32}\mathrm{S}$, and $^{36}\mathrm{Ar}$. The masses of $^{20}\mathrm{Na}$ and $^{32}\mathrm{Cl}$ are found to be in good agreement with the values from the 2003 Atomic Mass Evaluation (AME03) [G. Audi, A. H. Wapstra, and C. Thibault, Nucl. Phys. A 729, 337 (2003)], and the precision has been improved by a factor of 6 in both cases. The masses of $^{24}\mathrm{Al}$ and $^{28}\mathrm{P}$ are found to be higher than the values from AME03 by 9.5 keV ($3.2\ensuremath{\sigma}$) and 11.5 keV ($3.6\ensuremath{\sigma}$), respectively, and the precision has been improved by a factor of 2.5 in both cases. The new $^{32}\mathrm{Cl}$ mass is used together with the excitation energy of its lowest $T=2$ level and the mass of $^{32}\mathrm{Ar}$ to derive an improved superallowed ${Q}_{\mathrm{EC}}$ value of 6087.3(22) keV for this case. The effects on quantities related to standard-model tests including the $\ensuremath{\beta}$-$\ensuremath{\nu}$ correlation coefficient $a$ and the isospin-symmetry-breaking correction ${\ensuremath{\delta}}_{C}$ are examined for the $A=32$ case.