We propose and analyze an $H^2$-conforming virtual element method (VEM) for the simplest linear elliptic PDEs in nondivergence form with Cordes coefficients. The VEM hinges on a hierarchical construction valid for any dimension $d \geq 2$. The analysis relies on the continuous Miranda-Talenti estimate for convex domains $\Omega$ and is rather elementary. We prove stability and error estimates in $H^2(\Omega)$, including the effect of quadrature, under minimal regularity of the data. Numerical experiments illustrate the interplay of coefficient regularity and convergence rates in $H^2(\Omega)$.