Generally vector fields on manifolds are represented by derivatives of the form
$$\mathscr D : C^\infty(M) \to C^\infty(M)$$
where the vector field $V$ will act in the simple way
$$V(f) = \sum_{i=1}^n V^i \frac{\partial}{\partial x^i} f \circ \phi^{-1}(x)$$
I have seen in some posts that this definition may be in trouble for non-Hausdorff manifold, for instance
For me, a basic example of the beauty of this function-theoretic approach is the definition of a vector field as a derivation $D:C^\infty (M) \to C^\infty(M)$. The proof that such a derivation defines a vector field hinges upon the fact that $Df$ near a point $p$ only depends on $f$ near the point $p$. To prove this fact you use the fineness of your sheaf $\mathcal O_X$, i.e. the existence of partitions of unity. (It is true though that the failure of fineness in the non-Hausdorff case is of a different sort and might not break this particular theorem.)
On the other hand, I have seen a fair amount of authors also talk about vector fields on non-Hausdorff manifolds without any issue, like Hicks or Reeb. Derivatives on the splitting real line also seems to pose no particular issue, in particular.
Is there any potential problem in defining vector fields in such a way, linked perhaps to the non-uniqueness of limits in such spaces? It doesn't seem to cause any issue for the splitting real line, but consider for instance the complete feather, which has a branching point at every point of the manifold. Would there be an issue for instance of derivatives on different charts of the manifold having different values, or any other problem? Or is that definition of vector fields well defined even for such manifolds.