The answer appears to be as follows. ("Appears" because I haven't yet written out a detailed proof.)
Let $H^{[k]}_n(x)$ denote the variant of Hermite polynomials which are orthogonal with respect to the measure
$$ \frac{1}{\sqrt{2\pi k}}e^{-x^2/2k}dx . $$
Since the measure in the question is a product of the above measures (over all positive integers $k$), we have a family of orthogonal multivariable polynomials
$$ H^{[1]}_{n_1}(x_1) H^{[2]}_{n_2}(x_2) \cdots H^{[j]}_{n_j}(x_j) ,$$
indexed by tuples $(n_1,\ldots,n_l)$. The orthogonal polynomials of the questions are linear combinations of these. More specifically, let $N = \sum_i i\cdot n_i$. Think of $(n_1,\ldots,n_l)$ as encoding a conjugacy class in the symmetric group $S_N$, where $n_i$ is the number of $i$-cycles in a permutation. We can use the character table of $S_N$ to change basis from conjugacy-class-bump-functions to characters-of-representations. Applying an analogous change of basis to the above products of Hermite polynomials (separately for each $N$) yields the polynomials described in the question. There are some normalization factors I have not mentioned, related to the fact that some of the bases mentioned above are orthogonal but not orthonormal.
Thanks again to Suvrit and John Wiltshire-Gordon for pointing me in the right direction. I have not yet looked into Richard Borcherds' suggestion that these might also be specializations of Macdonald polynomials.
