In this paper, we lay the fundations of the theory of Witt-Frobenius modules and of Witt vector bundles. It is the main geometric tool in a series of papers by the first two authors in which they prove the mod $p^2$ liftability of mod $p$ Galois representations of all fields and of all dimensions, as well as provide a brand new proof of the Norm Residue Isomorphism Theorem. Let $p$ be a prime, and let $S$ be a scheme of characteristic $p$. Let $r \geq 2$ be an integer. Denote by $W_r(S)$ the scheme of Witt vectors of length $r$, built out of $S$. The main objective of this paper concerns the question of extending (=lifting) vector bundles on $S$, to vector bundles on $W_r(S)$. We first study two elementary but significant cases, in which the answer is positive: line bundles, and the tautological vector bundle of a projective bundle over an affine base. We give several applications, showcasing the systematic use of the Teichm\"uller lift of line bundles. First, we offer a simple formulation of classical results in deformation theory of smooth varieties over a field $k$ of characteristic $p$ and we extend them to reduced $k$-schemes. These results were recently recovered, in another form, by Stefan Schr\"oer. Then, we show that the tautological vector bundle of the Grassmannian $Gr_{\mathbb{F}_p}(m,n)$ does not extend to $W_2(Gr_{\mathbb{F}_p}(m,n))$, if $2 \leq m \leq n-2$. This permits to recover, and generalize, a recent result of Maciej Zdanowicz about the non-liftability of some projective bundles. In the Appendix, we give algebraic details on our new approach to Witt vectors, using divided powers and polynomial laws. In particular, we produce a natural surjection $W_2 \circ W_r \longrightarrow W_{r+1}$, of ring schemes over $\mathbb{F}_p$.