(Too long for the comment field.)
I understand that the question is not whether such functions exist (existence is easy to show) but how one can be produced explicitly.
It seems reasonable to take the ansatz $f(z)=g(|z|)$ where $g$ is a function on $[0,1]$ which is to be determined. The orthogonality relation is then recast in terms of real polynomials $p_n(r)=\int_0^{2\pi} |He_n(re^{it}|^2 r dt$. Precisely, $g$ must be orthogonal to $p_n$, $n>1$ on the positive halfline.Observe that the coefficients of $p_n$ are the squares of coefficients of $He_n$, only attached to different degrees (all odd). It is natural to suspect that $p_n$ or perhaps the polynomials $q_n$ defined by $p_n(x)=xq_n(x^2)$, are related to the Laguerre polynomials. If true, this would help with orthogonality but I could not see such a relation.