Manifolds $Mfld$ and (all) G-bundles $Bndl_G$ (on arbitrary manifolds) are both (very) nice objects of the topoi associated the following respective sites:
- Euclidean balls + $C^{\infty}$ structure.
- The above + $G$-structure.
You should be a bit careful in the wording of your description of manifold: the chart is not part of the data! But realizing this is not pedantry, it leads to the observation that I think you want to get at: the cover gives you a presentation of a `replacement' of your manifold. (Think of this in terms of (co)fibrant replacement in model categories.)
Also, manifolds themselves, often can be constructed in a very similar fashion to presenting a vector bundle by a map $M \to BG$, where we think of this as a simplicial map from the simplicial object constructed from a cover to the simplicial version of $BG$ (rather than a stack). For example, say a manifold $M$ fibers over a base manifold $X$ into smooth fibers $N$, i.e., we have a fiber diagram $N \to M \to X$. Then $M$ is realized via pullback along a map from $X$ to the classifying stack $BDiff(N)$; and here of course one can use charts, etc., to convert to simplicial objects (which, at least for me, helps with the visualization).
Also helpful: if one considers the $\infty$-category of $\infty$-stacks (algebraic or smooth), then the simplicial versions ($\cdots G \to *$) of $BG$ coincide with its stacky versions (the functor $BG(X) =~ ${stuff on $X$}). The idea is clear: a 1-stack, in particular $BG$, is a sheaf of 1-types, and these can always be viewed as homotopy types (i.e., $\infty$-types). The simplicial version of $BG$ is just choosing a presentation.