Is there a notion of "space" such that vector bundles can be understood in this way?
As Qiaochu said, probably what you want are topological stacks.
Let $T$ be a small full subcategory of $\mathrm{Top}$, with the Grothendieck topology of open covers, and consider the 2-category of stacks of groupoids on $T$. Call its objects "spaces" and its morphisms "continuous maps". You could call its 2-cells "continuous transformations".
Any topological space $X$ determines a sheaf $\mathrm{Top}(-,X)$ on $T$ and hence a stack. This functor is fully faithful on $T$, and often on a much larger subcategory of $\mathrm{Top}$. (For instance, if $T=\{\mathbb{R}^n\}$ then the functor is fully faithful on at least all topological manifolds.)
The functor $T^{op} \to \mathrm{Gpd}$ defined by sending $X\in T$ to the groupoid of real vector bundles over $T$ is a stack, because vector bundles can be glued together over open covers. Call it your $\mathbb{R}\mathbf{Mod}$.
If $X\in T$ (and often for many more $X\in\mathrm{Top}$), then by the Yoneda lemma, the groupoid of continuous maps and continuous transformations $\mathrm{Top}(-,X) \to \mathbb{R}\mathbf{Mod}$ is equivalent to $\mathbb{R}\mathbf{Mod}(X)$, i.e. the groupoid of real vector bundles on $X$.