Well, we do have the tangent bundle, and yes on there you would use tuples (p, X) to represent points, and in principle X could of course be typed as Atlases and charts · Manifolds.jl for cotangent vectors.
Beyond that I think we simply did not yet have the need nor any one formalising these aspects on manifolds much more. For my applications in optimization I for now always only needed tangent vectors; for the one case of the vector bundle newton we rephrased it (via sharp/flat) as a tangent bundle problem before implementing.
If you need more, it is probably best to either discuss that here or in a new issue in Manifolds.jl so we can help you implement / draft what you need.