Can you find [SO₃²⁻] with just the two acid dissociation constants Kₐ₁ and Kₐ₂? Sulfurous acid, $\ce{H2SO3}$, is a diprotic acid.
\begin{align}
K_{a1} &amp;= 1.6 \cdot 10^{-2}\\
K_{a2} &amp;= 6.4 \cdot 10^{-8}\\
\end{align}

Is it possible to find the concentration of $\ce{SO3^2-}$ from this?

I would think you would need to know the concentration of at least one thing. It is like trying to solve for 3 unknowns with 2 equations!
No, it is not possible.

For example, as total concentration of acid approaches zero, concentration of sulfate would also approach zero.