Is there a notion of "space" satisfying the following requirements?


1. Spaces form (at least) a category; morphisms between spaces are called "continuous maps."
2. Every topological space is a space, and the inclusion of $\mathbf{Top}$ in the category of spaces is fully faithful and injective on objects.
3. The category $\mathbb{R}\mathbf{Mod}$ is a space.
4. If $X$ is a topological space, a vector bundle over $X$ can be described as a continuous map $X \rightarrow \mathbb{R}\mathbf{Mod}.$


(I don't mind if some of these requirements are partially violated, they're just meant to be guidelines.)



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.


1. 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".
2. 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.)
3. 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 $\mathbb{R}\mathbf{Mod}$.
4. 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 $X$.


