It appears very intuitive to me that, just as a tangent space can be constructed on every point on the manifold, a Hilbert space should also be implementable as it is also a vector space. This would mean that each "tangent function" would have its own transformation properties, and a connection associated so that we can parallel transport them. Does this make sense? How can this be done?

My attempt at going around this has been to inspire myself on the tangent bundle structure. I mean that just as the tangent bundle is $$TM \:\colon = \bigcup\limits_{x\in \mathcal{M}} \{x\} \times T_xM,$$ my idea is to have $\mathcal{H}$ as our fiber, the manifold $\mathcal{M}$ as the base space, and $\mathcal{M} \times \mathcal{H}$ as the total space, so that, in analogy, we can have a $\mathcal{H}_x$ at each point in the space.

To me this is important because then we can implement bra-kets on the spacetime, introduce creation and annihilation operators, and decompose the QFT functions into modes. If there is a way to do this without considering my question I would greatly appreciate if you could explain this.