I believe this is the best you can get: if $f\colon X\rightarrow Y$ is a continuous map between top. spaces and $\pi\colon E\rightarrow X$ is a fiber bundle, these together form a pullback diagram. The pullback exists and is known as the pullback bundle $f^{\ast}E\rightarrow X$.
Then, there is a natural map $\operatorname{Hom}(Y,E)\rightarrow\operatorname{Hom}(X,f^{\ast}E)$ ($\operatorname{Hom}$ is taken in the category of vector bundles, i.e. these are the global sections) that is induced by pulling back (in the sense of precomposition) sections along $f$.