There are many types of finite dimensional Cartan type Lie algebras: to recover all finite dimensional simple Lie algebras in characteristic $p>3$ one has to define them over arbitrary finite dimensional divided power algebras (not just over functions on Frobenius neighbourhoods) and then also consider their natural filtered deformations. Since you are interested in the the special case of Frobenius neighbourhoods , the algebras in question will be restricted (in general, they are not). In this case one knows basically everything about irreducible restricted representations and almost nothing about the non-restricted ones (one notable exception is the Witt algebra where all irreducibles are described by Chang in 1941). The case of nontrival $p$-characters is extremely difficult already for $W(2,{\underline 1})$ (which is the full derivation algebra of the truncated polynomial ring in two variables).

The description of restricted representations resembles Rudakov's description of graded
irreducible representations of Lie algebras of Cartan type over complex numbers: the majority of irreducibles are induced from irreducible modules over standard maximal
subalgebras (vector fields without constant terms) and the non-induced ones appear as composition factors of modules arising from the de Rham complex. In characteristic $p$, complications arise in types $H$ and $K$ when $p$ is small. This is due to the fact that fundamental Weyl modules over $Sp_{2n}$ can be reducible when $p\le n$. But one knows dimensions and can write down character formulae (this will of course rely on the modular Lusztig conjecture for $SL_n$ and $Sp_{2n}$).