 Solution

Take bundle charts ${\Phi }_i:E{|}_{U_i}\to U_i\times {\mathbb{R}}^r$ with transition functions $g_{ij}:U_i\cap U_j\to GL(r,\mathbb{R})$. Over $f^{-1}(U_i)$ defineTheir transition functions are $g_{ij}\circ f$, so they give the fibre producta smooth manifold and make its projection to $M$ a rank-$r$ pullback vector bundle.

A local section $s:U\to E$ determines the sectionof $f^{\ast }E$ over $f^{-1}(U)$. It is unique with second component $s\circ f$.