The best I can do at this point is give you some interpretation of the Hopf algebroids associated to $X(n)$.
We can't really give a complete description of the stacks associated to these Hopf algebroids for the simple reason that their homotopy groups are pretty uncomputable. However, we can say a little about the role these play in homotopy theory.
First, no matter what $R$ is, the Hopf algebroid in spectra $(R, R \wedge R)$ attempts to recover the sphere from its cobar construction, and if $R_*R$ is a flat $R_*$-module we get a Hopf algebroid $(R_*, R_* R)$ attempting to do the same. In the cases of $X(n)$ these spectral sequences converge, so in some sense we should think of $(X(n)_*, X(n)_* X(n))$ as just another presentation of the moduli of formal groups.
In a little more detail, let's consider $MU_* X(n)$. This is a subring $MU_*[b_1, b_2, \ldots, b_n] \subset MU_* MU$. The latter ring parametrizes pairs of a formal group law together with a strict isomorphism $f(x) = \sum b_i x^{i+1}$ out to another formal group law; the former ring is the subring parametrizing a formal group law together with the truncation with a strict isomorphism out determined only up to the first $n$ coefficients. This is a natural description in terms of the $MU_* MU$-coaction, and so you might think of $X(n)$ as being associated to a moduli $\mathcal{M}_{fg}(n)$ of "formal groups equipped with a coordinate, determined up to the n'th stage".
Even further, you can consider the map of Hopf algebroids $(MU_*, MU_* MU) \to (MU_*, \pi_* (MU \wedge_{X(n)} MU))$. The latter computes the homotopy groups of $X(n)$ and is definitely the Hopf algebroid associated to the cover $Spec(MU) \to \mathcal{M}_{fg}(n)$.
The $X(n)$-based Adams spectral sequence is then something you might think of as being associated to a Cartan-Eilenberg type spectral sequence associated to an exact sequence of Hopf algebroids $$ (MU_* X(n), \pi_*(MU \wedge MU \wedge X(n))) \to (MU_*, MU_* MU) \to (MU_*, \pi_*(MU \wedge_{X(n)} MU)), $$ which in turn comes from considering a sequence of covering stacks $$ Spec(MU) \to \mathcal{M}_{fg}(n) \to \mathcal{M}_{fg}. $$