Consider a countable family of finite-rank vector bundles $V_k$ over a finite-dimensional smooth manifold $M$. The direct limit of such a family is still a topological vector bundle even though it may have infinite rank.

The first question is: Is there a natural structure of smooth manifold on the total space of $\varinjlim V_k$?

And the second one: Is it true, that the functor of smooth sections commutes with direct limit? I.e, is it true that $$\mathcal{C}^\infty(M,\varinjlim V_k) = \varinjlim \mathcal{C}^\infty(M,V_k)$$

Or turning things upside down -- is there a choice of smooth structure such that the above equation holds? If so, how does it look like?