One often times thinks of the Dieudonne module $M(X)$ of a $p$-divisible group (say over $k$, a perfect characteristic $p$ field) as being some sort of cohomology theory

$$M:\left\{p\text{- divisible groups}/k\right\}\to \left\{F\text{-crystals }/k\text{ with slopes in }[0,1]\right\}.$$

This is supported by the famous observation of Mazur-Messing-Oda that if $A/k$ is an abelian variety, then

$$M(A[p^\infty])=H^1_\text{crys}(A/W(k))$$

as $F$-crystals.

There are two natural questions that come from this, in my opinion. First, can one make sense of $M(X)$ as a crystalline cohomology group (in a literal sense) if $X$ 'comes from a scheme' (e.g. is the $p^\infty$-torsion of some group scheme). I've asked this here as well as asking when $M$ (an $F$-crystal) 'comes from a scheme'.

But, the other question, and the one I am asking here is the following. Is there some way to interpret the understanding of the Dieudonne module 'as cohomology' in a rigorous way? For example, is there a natural site over $X$ (perhaps with underlying category $p$-divisible groups over $X$) such that $M(X)$ is cohomology on this site? I would even be interested in knowing if there is a rigorous way of considering $M(X)/pM(X)$ as a cohomology theory (in the Mazur-Messing-Oda setup it's $H^1_\text{dR}(A/k)$).

The obvious guess is to create a crystalline/infinitesimal site on $X$, but I don't really know how to make this precise.