Bear in mind that I don't have a formal education in mathematics. If functions form an abstract vector space then a single function can be considered as a member of a vector space. Then how can we obtain a vector field with functions, i.e., an object that attaches a unique function at each point on a given space?. I am not even sure whether I say make sense or whether my logic is correct.
Essentially what I wonder is that whether one can form an object that would attach a function to a point in space uniquely, as in one function for each point like vector fields attaching vectors to each point in space.
A "vector field of functions", an object which attaches a unique function to every point in, say $\mathbb{R}^n$, would basically be another function $g : \mathbb{R}^n \to C(X,Y)$, where $C(X,Y)$ denotes the functions from some set $X$ to another set $Y$. That would be the most basic way to define such an object. To my knowledge, one of the most common ways a structure like this shows up in mathematics is in forms of sections of fibre bundles, specifically differential forms as sections of the cotangent bundle of a manifold. Might be kind of scary concepts if you've not had a formal education in mathematics, but I will try to give a few short explanations for why one wants to study these generalizations:
If the space to which you want to attach functions is not $\mathbb{R}^n$, but maybe a circle, a sphere, a torus, a Klein bottle, of course you will simply need to replace every instance of $\mathbb{R}^n$ with your space in question. However, these spaces are a little bit more difficult to handle compared to $\mathbb{R}^n$ in a few ways: A sphere (which we will denote by $S^n$ from now on) is not a vector space, so adding points on a sphere makes no sense, and as such, it will be difficult to define what it means for a function on a sphere to be differentiable. Why is this important to your question?: Most likely, you want the values of your vector field of functions $g : S^n \to C(X,Y)$ not to be completely arbitrary, but maybe to be related in some smooth manner. When the space in question is "not too bad", like the above mentioned examples, they're manifolds, and a notion of differentiability can be established.
Furthermore: Perhaps you don't want to attach the same kind of function to every point. Like, at point $x_1$ you only allow functions from the space $C_1$, but at point $x_2$ you only allow functions from the space $C_2$. Then you will have to modify your vector field to read $g : S^n \to \bigcup_i C_i$, where the union goes over all different function spaces. That is basically what sections of a fibre bundle are. A fibre bundle is a space, the base space (like our $S^n$) which has another space, the fibre, attached to every point of it (like our $C_i$), and a section of this bundle is a map which chooses an element of the attached space for every point on the base space.
I think the first nontrivial example of such a situation that most mathematicians first come across is differential forms: Those are functions which attach to every point $x$ on a manifold a function $\omega_x$, and this function takes tangent vectors $X_1,\dots,X_n$ on the manifold at that point and assigns a real number to them in a multilinear way. Note that one requires that the function $\omega_x$ only takes tangent vectors attached to the manifold at the point $x$, so here we really have a different function space at every point, because it takes different tangent vectors at every point.
Maybe that gives you some ideas on how your concept is being studied!