The following corresponds to exercise 1 in Sec 1, Chp. 4 of Hirsch's book Differential Topology.
Let $\xi,\eta$ be two vector bundles over the same base $B$. Assume that $B$ us paracompact and $A\subset B$ is closed. Then, every bundle map $f:\xi|_A \rightarrow \eta|_A$ covering $id_A$ extends to a bundle morphism $f:\xi|_A \rightarrow \eta|_A$ covering $id_W$, for some neighbourhood $W$ of $A$.
My idea is to use the fact that the base is paracompact to apply Tietze extension theorem: If $X$ is a normal topological space and $A\subset X$ is closed, then every continuous function $f:A\rightarrow \mathbb R$ extends to a function $\tilde f:X\rightarrow \mathbb R$.
Let $\{(U_\alpha,\varphi_\alpha)\}_{\alpha\in \Lambda}$ and $\{(V_\beta,\psi_\beta)\}_{\beta\in \Xi}$ be a family of vector bundle charts for $\xi$ and $\eta$ covering $A$. By taking the intersections if necessary and restricting the charts, we can assume that $\Lambda=\Xi$ and that $U_\alpha=V_\alpha$.
Let $F_{\alpha\beta}$ denote the induced map by $\varphi_\alpha|^{-1}_{E|_{A\cap U_\alpha}}f \psi_\beta|^{-1}_{F|_{A\cap U_\beta}} : U_\alpha\cap U_\beta \times \mathbb R^n \rightarrow U_\alpha\cap U_\beta \times \mathbb R^m$ given by
$$ (x,\vec v) \longmapsto (x,F_{\alpha\beta}(\vec v)) . $$
Fix $\vec v\in\mathbb R^n$ and define $m$ maps
$$ \begin{array}{rcl} F_{\vec v}^j :A & \longrightarrow & \mathbb R \\ x & \longmapsto & \bigg (F_{\alpha\beta}(\vec v)\bigg)^j, \end{array} $$
where $x$ belongs to $U_\alpha\cap U_\beta$ (here $\alpha$ can be equal to $\beta$). We can use a partition of unity to fix the definitions on the overlaps $U_\alpha\cap U_\beta\cap U_{\alpha'}\cap U_{\beta'}$ and get well-defined functions on $A$.
Using Tietze Extension Theorem, each $F_{\vec v}^j$ extends to a neighbourhood $W_{\vec v}^j$ of $A$ (let me use the same symbols to denote the extension). Let $W_{\vec v}$ be their intersection and set
$$ \begin{array}{rcl} F_{\vec v} :W_{\vec v} & \longrightarrow & \mathbb R^m, \\ x & \longmapsto & \bigg (F_{\alpha\beta}(\vec v)^1,\dots,F_{\alpha\beta}(\vec v)^m\bigg). \end{array} $$
Now, if $A$ is compact, the cover $\{W_{\vec v}\}_{\vec v\in\mathbb R^n}$ admits a finite subcover. Define $W$ as its intersection and put
$$ V = W \cap \left(\bigcup_{\alpha\in A} U_\alpha \right). $$
Finally, I define $\tilde f: \xi|_V \rightarrow \eta|_V$ as the bundle morphism whose local representations are
$$ (x,\vec v) \longmapsto (x,F_{\vec v}|_{U_\alpha\cap U_\beta}(x)) . $$
I am not sure if my proof is correct, even assuming compactness. And even if it is, I find it rather complicated. The exercise in the book does not present an asterisk, which means that the exercise should not be really demanding.
Hence, is my proof correct (assuming compactness)? Is there a simpler proof (for the compact case)? And how can I prove the result when $A$ is not assumed to be compact?