A contact structure on $M^{2n+1}$ defines a $G$-structure (actually, it defines more than one, but there is a 'minimal' $G$-structure that is preserved by all contact transformations, and that is the one that people usually consider).  Conversely, a $G$-structure on $M^{2n+1}$ comes from a contact structure provided that its intrinsic torsion satisfies the appropriate identities.

For example, a contact structure on $M$ is defined by a $2n$-plane field $D\subset TM$ (satisfying some nondegeneracy property).  So if $G_0\subset\mathrm{GL}(2n{+}1,\mathbb{R})$ is the linear subgroup that preserves the subspace $\mathbb{R}^{2n}\subset\mathbb{R}^{2n{+}1}$, then a contact structure defines a $G_0$-structure on $M$ and, conversely, that $G_0$-structure contains the information needed to recover $D$.  In that sense, the $G_0$-structure and the $2n$-plane field $D$ are equivalent information.

However, you will note that not every $G_0$-structure on $M$ defines a contact structure, because the corresponding hyperplane field defined by $G_0$ might be integrable, say. 

Now, because a contact structure $D\subset TM$ defines a linear bundle map $B:\Lambda^2(D)\to TM/D$ that is non-degenerate, there is a natural subgroup $G_1\subset G_0$ which is the subgroup that preserves a non-degenerate mapping $b:\Lambda^2(\mathbb{R}^{2n})\to\mathbb{R}^{2n+1}/\mathbb{R}^{2n}$, so the contact structure $D$ actually defines a canonical $G_1$-structure on $M$ in the obvious way.  Conversely, a $G_1$-structure on $M$ whose canonically induced mapping $B:\Lambda^2(D)\to TM/D$ coincides with the one induced by the 'reduced' Lie bracket on $M$ contains all the information in the contact structure and nothing else; automorphisms of the $G_1$-structure are exactly the contact transformations.  Moreover, there is no proper subgroup $G_2\subset G_1$ for which contact transformations of $M$ preserve some $G_2$-structure on $M$.  In this sense, the $G_1$-structure is the 'minimal' $G$-structure defined by a contact structure.