I want to know how to define the continuity of a pointwise orientation on a manifold in a specific way (if this is possible), analogously to the continuity of a vector field as a continuous map from the manifold to the tangent bundle. By a pointwise orientation on a manifold $M$ I mean a function that assigns to each point $p \in M$ an orientation of the tangent space $T_p M$.
The standard way I know of defining the continuity of this orientation is to state that at each point there is a chart that preserves orientation according to the pointwise orientation, that is, takes bases with the same orientation in $M$ to bases with the same orientation in $\mathbb R^d$.
There is a similar definition of continuity of vector fields, in which we define a vector field as a function that assigns to each point $p \in M$ a tangent vector in $T_p M$, and we say that this vector field is continuous according to it's continuity in charts. Nevertheless, we can ignore this definition of continuity via charts and say that a vector field is continuous if it is a continuous function $X\colon M \to TM$, considering the topology of $TM$ as the tangent bundle. So what I want to know is:
Is there a way to assign some kind of topological structure to the space of orientations of tangent spaces $T_p M$ in such a way that the continuity of a pointwise orientation is just continuity with respect to this topological structure?
I may also add that I understand an orientation of a linear space as an equivalence class of ordered bases of that space, or even better as an injective function from this equivalence class set to the set $\{1,-1\}$, but I don't know how to 'glue' these structures together for each tangent space $T_p M$ in such a way that there is some kind of fiber bundle structure and the (continuous) orientation of a manifold can be viewed as a kind of section of this bundle.
Given a smooth manifold, you can consider the cotangent bundle on the manifold (often denoted $T^\ast M$) which is given by point wise taking the dual of the tangent space. So to answer your last question, the “gluing” comes from the gluing you do on the tangent bundle. Now, we can consider exterior powers of the cotangent bundle. In particular, if the manifold is of dimension $n$, then we can pointwise look at the $n$’th exterior power of $T^\ast_p M$. This yields an $\mathbb{R}$-bundle over the manifold. If we remove the zero section (I.e pointwise remove $0$ from the fibre), then we are left with an $(\mathbb{R}-0)$-bundle over $M$. This can have one or two components. If it has one, then the manifold is non-orientable. If it has two, the manifold is orientable.
Now, I’m not entirely sure what you mean by a continuous local orientation, but a consistent choice of local orientation (which you need to define a global orientation) means picking out a continuous section of this bundle. Continuity automatically implies that we pick out only one component of the manifold, so you can also rephrase this as a consistent choice is just a choice of a component of this bundle. To answer your question, this topological space is the space of orientations.
If you’ve seen differential forms, notice that a continuous section is exactly a (continuous but not necessarily smooth) non-vanishing top-dimensional differential form, which is another common way of taking orientations on a manifold. Such a differential form gives an ordering of the basis of tangent vectors at each point. So you can see how all these definitions are compatible.