Holomorphic vector fields give elements of the first Hochschild cohomology group of the structure sheaf $$HH^1(O_X,O_X)\simeq H^1(X,O_X)\oplus H^0(X,T_X)$$ (via an easy part of the Hochschild-Kostant-Rosenberg theorem).
On the other hand the Hochschild cohomology may be identified (or indeed defined as) self-Ext of the identity functor on the category of coherent sheaves. Hence, by definition of self-ext of the identity, there's a functorial map $$HH^1(O_X,O_X)=Ext^1(Id_{O_X-mod})\longrightarrow Ext^1(F,F)$$ for any coherent sheaf $F$. This is one fancy way to express your construction. Likewise the second Hochschild cohomology gives deformations of the (derived) category of sheaves, and the corresponding element in $Ext^2(F,F)$ is the obstruction for $F$ to deform.
EDIT: Here's a little more to explain the compatibility with Torsten's answer. Any sheaf $F$ has a canonical extension by $F\otimes \Omega^1$, given by 1-jets of sections of $F$ $$0\to F\otimes \Omega^1 \to J^1(F) \to F\to 0.$$ We can contract this extension against a vector field to get an $Ext^1$ of $F$ by itself as desired. The functor taking $F$ to its extension is represented (as an integral transform) by the first-order neighborhood of the diagonal, i.e. by the 1-jet of the identity functor (which again can be contracted against any vector field). This is explicitly the map from vector fields to $Ext^1$ of the identity functor, i.e. Hochschild $HH^1$.
This universal extension can be rewritten as an element $$Hom(T_X[-1]\otimes F,F),$$ which in fact defines a homotopy Lie algebra structure on the shifted tangent sheaf $T_X[-1]$ (the Atiyah bracket). The sheaf Hochschild cohomology can then be identified as the universal enveloping algebra of this homotopy Lie algebra. So in some precise sense this universal action of vector fields (as Torsten describes) IS Hochschild cohomology, acting as the derived center of the category.