  Let $\mathscr{V}\mathrm{ect}_n$ be the moduli stack of vector bundles of rank
$n$ on schemes. We prove that, if $E$ is a Zariski sheaf of ring spectra which
is equipped with finite quasi-smooth transfers and satisfies the projective
bundle formula, then $E^*(\mathscr{V}\mathrm{ect}_{n,S})$ is freely generated
by Chern classes $c_1,\dotsc,c_n$ over $E^*(S)$ for any scheme $S$. Examples
include all multiplicative localizing invariants.
