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.