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Okay so $X$, viewed the functor $X(S)=\{\textrm{Maps}:S\to X\}$ has a natural map $\pi:X\to X_{dR}$, by pre-composing with $S_{red} \to S$.

There is a scheme $Aut_X$ of infinite type over a smooth $X$ whose set of points consists of pairs $(x\in X, t_x: X_{(x)}\cong \hat{D_n})$, where $\hat{D_n}$ is the formal disc, $\mathcal{O}(\hat{D_n}):=\mathbb{C}[[x_1,\ldots,x_n]]$. The fiber over each point $x\in X$ is the group $Aut(X_{(x)})$ of automorphisms of the formal neighborhood of $x$ preserving the maximal ideal, which is in turn (non-canonically) isomorphic to $Aut^0(\hat{D_n})$, the group of automorphisms of $\hat{D_n}$ preserving the origin.

Better $M$ has an action of the operators $\partial_i$, and we can exponentiate the action of all of $W_n$ to the group $\hat G$. That means that we can instead construct the associated bundle for the $\hat G$ torsor $Aut_X\to X_{dR}$, which will give us a bundle over $X_{dR}$ with fiber $M$ again. Now, we can pullback this bundle via $\pi$ to get a bundle on $X$ with fiber $M$. This time, however, it's a pullback of a sheaf of vector spaces on the deRham stack, which by (some people's definition) is a crystal of vector spaces on $X$.

Okay so $X$, viewed as the functor $X(S)=\{\textrm{Maps}:S\to X\}$ has a natural map $\pi:X\to X_{dR}$, by pre-composing with $S_{red} \to S$.

There is a scheme $Aut_X$ of infinite type over a smooth $X$ whose set of points consists of pairs $(x\in X, t_x: X_{(x)}\cong \hat{D_n})$, where $\hat{D_n}$ is the formal disc, $\mathcal{O}(\hat{D_n}):=\mathbb{C}[[x_1,\ldots,x_n]]$. The fiber over each point $x\in X$ is the group $Aut^0(X_{(x)})$ of automorphisms of the formal neighborhood of $x$ preserving the maximal ideal, which is in turn (non-canonically) isomorphic to $Aut^0(\hat{D_n})$, the group of automorphisms of $\hat{D_n}$ preserving the origin.

Better $M$ has an action of the operators $\partial_i$, and we can exponentiate the action of all of $W_n$ to the group $\hat G$. That means that we can instead construct the associated bundle for the $\hat G$ torsor $Aut_X\to X_{dR}$, which will give us a bundle over $X_{dR}$ with fiber $M$ again. Now, we can pullback this bundle via $\pi$ to get a bundle on $X$ with fiber $M$. This time, however, it's a pullback of a sheaf of vector spaces on the deRham stack, which by (some people's) definition is a crystal of vector spaces on $X$.

There is a scheme $Aut_X$ of infinite type over a smooth $X$ whose set of points consists of pairs $(x\in X, t_x: X_{(x)}\cong \hat{D_n})$, where $\hat{D_n}$ is the formal disc, $\mathcal{O}(\hat{D_n}):=\mathbb{C}[[x_1,\ldots,x_n]]$.

[edit in process, sorry.]

Second, in geometry: The module $M$ is a Harish-Chandra module for the pair $(W_n,Aut_0(C[[x_1,...,x_n]]))$. This means: $W_n$ has a Lie sub algebra $W^0$ of vector fields which vanish at the origin (so they have no constant vector field terms like $\partial_i$). $W^0$ (or perhaps its completion w.r.t order of vanishing at origin) is the Lie algebra of $Aut_0(C[[x_1,...,x_n]])$, as it consists of derivations which preserve the maximal ideal (not sure exactly how you make precise that it is "the Lie algebra", but anyways, there is an exponential map turning an integrable $W^0$ module into an $Aut_0(C[[x_1,...,x_n]]$)-module ). The assumptions on $M$ were precisely those that make $M$ an integrable $W^0$ module, and so we can regard $M$ as a $G$-module. Well now we have an associated bundle construction for the $G$-torsor $X\to X_{dR}$, and we can use this to produce a sheaf of vector spaces (not quasi-coherent!) over $X_{dR}$ with fiber $M$. We can pull back $M$ by $\pi$ to a sheaf $M_X$ (again, not quasi-coherent!) on $X$.

There is one more detail which is that $M$ wasn't only a $G$-module, but it also had the constant vector fields $\partial_i$. It turns out that these equip the sheaf $M_X$ with a crystal structure, as we can identify infinitesimally nearby fibers of $M_X$ via these fields. This I understand less clearly, although if one is comfortable with this technology it's probably clear why.

There is a scheme $Aut_X$ of infinite type over a smooth $X$ whose set of points consists of pairs $(x\in X, t_x: X_{(x)}\cong \hat{D_n})$, where $\hat{D_n}$ is the formal disc, $\mathcal{O}(\hat{D_n}):=\mathbb{C}[[x_1,\ldots,x_n]]$. The fiber over each point $x\in X$ is the group $Aut(X_{(x)})$ of automorphisms of the formal neighborhood of $x$ preserving the maximal ideal, which is in turn (non-canonically) isomorphic to $Aut^0(\hat{D_n})$, the group of automorphisms of $\hat{D_n}$ preserving the origin.

Now, the scheme $Aut_X$ is actually a $\hat G$-torsor over $X_{dR}$, where (I gather from David Ben-Zvi's comments) $\hat G$ is something more like an inductive limit of the groups of automorphisms on $n$th infinitessimal neighboroods of the origin in $D_n$.

Second, in geometry: The module $M$ is a Harish-Chandra module for the pair $(W_n,Aut_0(C[[x_1,...,x_n]]))$. This means: $W_n$ has a Lie sub algebra $W^0$ of vector fields which vanish at the origin (so they have no constant vector field terms like $\partial_i$). $W^0$ (or perhaps its completion w.r.t order of vanishing at origin) is the Lie algebra of $Aut_0(C[[x_1,...,x_n]])$, as it consists of derivations which preserve the maximal ideal (not sure exactly how you make precise that it is "the Lie algebra", but anyways, there is an exponential map turning an integrable $W^0$ module into an $Aut_0(C[[x_1,...,x_n]]$)-module ). The assumptions on $M$ were precisely those that make $M$ an integrable $W^0$ module, and so we can regard $M$ as a $G$-module. Well now we have an associated bundle construction for the $G$-torsor $Aut_X\to X$, and we can use this to produce a sheaf of vector spaces (not quasi-coherent!) over $X$ with fiber $M$.

Better $M$ has an action of the operators $\partial_i$, and we can exponentiate the action of all of $W_n$ to the group $\hat G$. That means that we can instead construct the associated bundle for the $\hat G$ torsor $Aut_X\to X_{dR}$, which will give us a bundle over $X_{dR}$ with fiber $M$ again. Now, we can pullback this bundle via $\pi$ to get a bundle on $X$ with fiber $M$. This time, however, it's a pullback of a sheaf of vector spaces on the deRham stack, which by (some people's definition) is a crystal of vector spaces on $X$.

I am writing to post an answer to my own question. The answer below consists of what I was able to jot down from a seminar talk by Jacob Lurie, and a patient followup explanation by Roman Travkin. Of course, mistakes and naivete in the translation are solely attributed to me. Since the answer was given to me in response to my asking on MO, it seems that karma dictates that I record it here.

Claim: Suppose $X$ is smooth, and over $\mathbb{C}$. Then $\pi$ defines a $G$-torsor over $X_{dR}$, with total space $X$, and $G$ here is $Aut_0(C[[x_1,...,x_n]])$ where $X$ is of dimension $n$.

More naturally, the preimage $\pi^{-1}(\pi(x))$, for $x \in X$, is $Aut_0 X_{(x)}$, the automorphisms of the formal neighborhood $X_{(x)}$, which preserve the maximal ideal $(m_x)$.

I am writing to post an answer to my own question. The answer below consists of what I was able to jot down from a seminar talk by Jacob Lurie, and a patient followup explanation by Roman Travkin, followed by a correction by David Ben-Zvi. Of course, mistakes and naivete in the translation are solely attributed to me. Since the answer was given to me in response to my asking on MO, it seems that karma dictates that I record it here.

There is a scheme $Aut_X$ of infinite type over a smooth $X$ whose set of points consists of pairs $(x\in X, t_x: X_{(x)}\cong \hat{D_n})$, where $\hat{D_n}$ is the formal disc, $\mathcal{O}(\hat{D_n}):=\mathbb{C}[[x_1,\ldots,x_n]]$.

[edit in process, sorry.]