  We identify Whittaker vectors for $\mathcal{W}_k(\mathfrak{g})$-modules with
partition functions of higher Airy structures. This implies that Gaiotto
vectors, describing the fundamental class in the equivariant cohomology of a
suitable compactification of the moduli space of $G$-bundles over
$\mathbb{P}^2$ for $G$ a complex simple Lie group, can be computed by a
non-commutative version of the Chekhov-Eynard-Orantin topological recursion. We
formulate the connection to higher Airy structures for Gaiotto vectors of type
A, B, C, and D, and explicitly construct the topological recursion for type A
(at arbitrary level) and type B (at self-dual level). On the physics side, it
means that the Nekrasov partition function for pure $\mathcal{N} = 2$
four-dimensional supersymmetric gauge theories can be accessed by topological
recursion methods.
