I believe that if $X$ is a normal variety over a perfect field $k$, there is an obstruction theory for $X$ with values in $\mathrm{Ext}^2(\Omega_X, \mathcal O_X)$ (one should not talk about "the" obstruction space).

Let $L_X$ be the cotangent complex of $X$ over $k$; the canonical obstruction theory of $X$ over $k$ has values in $\mathrm{Ext}^2(L_X, \mathcal O_X)$. If we denote by $\underline H^j(L_X)$ the $j^{\rm th}$ cohomology sheaf of $L_X$, we have $\underline H^j(L_X) = 0$ for $j > 0$, $\underline H^0(L_X) = \Omega_X$, while for $j < 0$ the sheaf $\underline H^j(L_X)$ is supported in the singular locus of $X$; hence $\mathrm{Ext}^i(\underline H^j(L_X), \mathcal O_X) = 0$ for $i =0 $ and $i = 1$, because $X$ is $S_2$ and the codimension of the singular locus is at least $2$.

I claim that the projection $L_X \to \Omega_X$ induces an isomorphism of $\mathrm{Ext}^2(\Omega_X, \mathcal O_X)$ with $\mathrm{Ext}^2(L_X, \mathcal O_X)$. This follows easily from the existence of the spectral sequence $$ E_2^{ij} = \mathrm{Ext}^i(\underline H^{-j}(L_X), \mathcal O_X) \Longrightarrow \mathrm{Ext}^{i+j}(L_X, \mathcal O_X) $$ together with the vanishing of the terms $E_2^{ij}$ for $j > 0$, $i=0$ or $1$.

I believe that if $X$ is a normal variety over a perfect field $k$, there is an obstruction theory for $X$ with values in $\mathrm{Ext}^2(\Omega_X, \mathcal O_X)$ (one should not talk about "the" obstruction space).

Let $L_X$ be the cotangent complex of $X$ over $k$; the canonical obstruction theory of $X$ over $k$ has values in $\mathrm{Ext}^2(L_X, \mathcal O_X)$. If we denote by $\underline H^j(L_X)$ the $j^{\rm th}$ cohomology sheaf of $L_X$, we have $\underline H^j(L_X) = 0$ for $j > 0$, $\underline H^0(L_X) = \Omega_X$, while for $j < 0$ the sheaf $\underline H^j(L_X)$ is supported in the singular locus of $X$; hence $\mathrm{Ext}^i(\underline H^j(L_X), \mathcal O_X) = 0$ for $i =0 $ and $i = 1$, because $X$ is $S_2$ and the codimension of the singular locus is at least $2$ (because $L_X = \Omega_X$ when $X$ is smooth, and regular implies smooth over a perfect field).

I claim that the projection $L_X \to \Omega_X$ induces an isomorphism of $\mathrm{Ext}^2(\Omega_X, \mathcal O_X)$ with $\mathrm{Ext}^2(L_X, \mathcal O_X)$. This follows easily from the existence of the spectral sequence $$ E_2^{ij} = \mathrm{Ext}^i(\underline H^{-j}(L_X), \mathcal O_X) \Longrightarrow \mathrm{Ext}^{i+j}(L_X, \mathcal O_X) $$ together with the vanishing of the terms $E_2^{ij}$ for $j > 0$, $i=0$ or $1$.