  We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exact
categories of vector bundles over monoid schemes. Our main results are the
complete description of the algebraic $K$-theory space of an integral monoid
scheme $X$ in terms of its Picard group $\operatorname{Pic}(X)$ and pointed
monoid of regular functions $\Gamma(X, \mathcal{O}_X)$ and a description of the
Grothendieck-Witt space of $X$ in terms of an additional involution on
$\operatorname{Pic}(X)$. We also prove space-level projective bundle formulae
in both settings.
