Let me parse your question on three levels.

First, there's an obvious counterpart to W-algebras in the p-adic world that you allude to. Namely, "W-algebras = endomorphisms of Whittaker models". More precisely, for any $\rho:SL_2\to G$ we can construct a generalized Whittaker model $W_\rho$ as $(N_\rho,\psi_\rho)$-invariant functions on $G$, where $N_\rho$ is the "upper triangular" unipotent subgroup built out of $\rho$ and $\psi_\rho$ is its character determined by the lowering operator of the $SL_2$. The corresponding $W$-algebra is its endomorphism algebra, which like endomorphisms of any induced representation has a abstract description that you can call "quantum Hamiltonian reduction" (in this case it gets "scarier" names semi-infinite cohomology or specifically Drinfeld-Sokolov reduction). So these are very fundamental objects in p-adic representation theory (best known for the "usual" Whittaker model, i.e. when $\rho$ is principal, in which case this algebra is famously commutative).

But I think you're asking about the more novel features of W-algebras in the world of vertex / chiral algebras etc. These don't have any obvious analog even in the t-adic world of representation theory over $k((t))$ - UNLESS you are in the setting of de Rham geometric Langlands, i.e. $k$ is the complex numbers and we're studying a different kind of representation theory -- not smooth reps of groups over local fields, but Lie algebra representations or (better) smooth CATEGORICAL reps of groups over local fields. The two main points here are

there's a "level" parameter, coming from the Kac-Moody central extension (or alternatively the "quantization" parameter for quantum groups). The corresponding (Lie algebraic) theory of W-algebras gets really rich as we vary this parameter. This "quantum" parameter does not currently have any known analog in the local Langlands program (t-adic or p-adic) [except for the rational level cases where q is a root of unity, which have to do with metaplectic analogs].

there's the theory of opers, which is a "semiclassical" version -- i.e. you replace Whittaker models by Kostant or Slodowy slices, but crucially you have to add in the Kac-Moody extension. This is absolutely crucial to de Rham geometric Langlands, in particular its recent solution (which then propagated to other versions of the conjecture), but again has no analog outside the de Rham setting.

That being said (and maybe finally getting to your actual question) it's very natural to imagine there will be a version of some of this de Rham theory, including W-algebras and opers, in the p-adic Langlands program, i.e. we don't study smooth representations of p-adic or t-adic groups but locally analytic ones (which in particular give interesting Lie algebra representations). This is really a great question, that I've heard experts discussing, but as far as I know there's nothing at all in the literature of this kind.