Are polynomials dense is L2 of the unit disk?

No. For instance, $g(z)=\overline{z}$ is orthogonal to every polynomial (proof sketch: if $f(z)=z^n$, then $\langle f,g\rangle=\int_D f\overline{g}=\int_D z^{n+1}=0$ by either direct computation in polar coordinates or using the symmetries of $z^{n+1}$). It follows that $g$ is not in the closure of the polynomials.