Are you sure it is about vector bundles?
Any topological vector space is contractible. Any locally convex topological vector space is locally contractible. This is straightforward from the definitions. The compact-open $C^\infty$ topology is defined by a family of semi-norms and hence locally convex.
In the case of arbitrary fiber bundles, the space of sections can indeed fail to be locally contractible. For example, consider the trivial bundle $\mathbb N\times\{0,1\}$ over $\mathbb N$ (they are 0-dimensional manifolds). The space of sections is not locally contractible (in fact, homeomorphic to the Cantor set). You can build connected examples as well, e.g. consider the trivial circle bundle over a surface with infinitely many handles.