In the affine, characteristic zero, smooth case, you can obtain $\mathcal D$ as the envelopping algebra of the Lie-Rinehart pair $(\mathcal O_X, \mathrm{Der}(X))$ (I think this is precise in for the global sections only, but I do not recall having seen the notion of a sheaffy Lie-Rinehart pair...).
In the general case, things are surely harder, since $\mathcal D$ is not generated in general from derivations and functions only: you need higher order generators. (In fact, a very famous conjecture of Nakai claims that an affine variety is smooth iff $\mathcal D$ is generated by functions and derivations.) In positive characteristic, things are worse, as you already need more generators on $\mathbb A^1$.
PS: I cannot find Nakai's Higher order derivations, I in MathSciNet not nor in Zentralblatt... I don't think lots of progress has happened with respect to that conjecture, sadly (Villamayor-Mount, Becker, &c have special cases)
