We perform the first search for an isotropic non-tensorial gravitational-wave background (GWB) allowed in general metric theories of gravity in the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) 12.5-year data set. By modeling the GWB as a power-law spectrum, we find strong Bayesian evidence for a spatially correlated process with scalar transverse (ST) correlations whose Bayes factor versus the spatially uncorrelated common-spectrum process is $99pm 7$, but no statistically significant evidence for the tensor transverse, vector longitudinal and scalar longitudinal polarization modes. The median and the $90%$ equal-tail amplitudes of ST mode are $mathcal{A}_{mathrm{ST}}= 1.06^{+0.35}_{-0.28} times 10^{-15}$, or equivalently the energy density parameter per logarithm frequency is $Omega_{mathrm{GW}}^{mathrm{ST}} = 1.54^{+1.20}_{-0.71} times 10^{-9}$, at frequency of 1/year.
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