As requested by user @AntonPetrunin, I am turning my comment into an answer.

That fails already for rank $1$ vector bundles over $M=\mathbb{R}\times \mathbb{S}^1$.  For the tautological rank $1$ vector bundle $F$ on $\mathbb{S}^1=\mathbb{RP}^1$, the pullback $E=\text{pr}_2^*F$ gives a counterexample.  For every compact subset $K$ of $\mathbb{R}\times \mathbb{S}^1$, the image $\text{pr}_1(K)$ is a compact subset of $\mathbb{R}$, hence bounded.  Thus, there exists $t\in \mathbb{R}\setminus \text{pr}_1(K)$.  For the section $\sigma_t:\mathbb{S}^1\to M$ by $\sigma_t(u)=(t,u)$, the pullback $\sigma_t^*E$ equals $F$.