I don't have a reference, but the following parametrization may help to describe this moduli space. For simplicity I consider polygons with marked vertices. For each polygon in your sense, there is a representation by the Schwarz-Christoffel formula
$$f(z)=C\int_a^z\prod_{k=1}^n (\zeta-a_k)^{\alpha_k-1}d\zeta,$$
where $f$ is an inverse Riemann map of the upper half-plane onto your polygon, $C\neq 0$ and $a$ are arbitrary constants; $\pi\alpha_j>0$ are the inner angles subject to
$$\alpha_1+\ldots+\alpha_n=n-2,$$
and $a_j$ are distinct points on the real line (pre-vertices). The polygons are equal if the corresponding $C$ and $a$ are equal and the sequences
$(a_1,\ldots,a_n)$ are mapped to each other by an element of $SL(2,R)$.
The description is simplified if you set one pre-vertex to be $\infty$. Then you have $n-1$ tuples $(a_1,\ldots,a_{n-1})$ modulo the action of
the affine group $x\mapsto cx+d$, and the sum of the angles, except one is $<n-2$.
Hyperbolic polygons have a similar description, but without a formula: to each sequence of angles and an
equivalence class of pre-vertices (images of vertices under a Riemann map) there exists exactly one polygon
and the angles are subject to the restriction
$$\sum_{j=1}^n\alpha_j< n-2.$$
This is a very old result going back to E. Picard. (The first modern proof is in MR0143901
Heins, Maurice
On a class of conformal metrics.
Nagoya Math. J. 21 (1962), 1–60.)
Putting everything together, the moduli space
of pairs (Euclidean polygon, vertex) is
$$\mathbb{C}^*\times \mathbb{C}\times\Delta_{n-1}\times\Delta_{n-3},$$
where $\Delta_k$ is an open simplex of dimension $k$. The first two multiples are for $C$ and $a$, the second for the angles and the third for pre-vertices. (Choosing a vertex allows to mark all vertices in the natural order.)
Similarly in the hyperbolic case it is
$$PSL(2,\mathbb{R})\times\Delta_n\times\Delta_{n-3},$$
where $PSL(2,\mathbb{R})$ is the group of hyperbolic motions.
Remarks.
The corresponding problem for spherical polygons is wide open. Only for triangles there is a complete description of the moduli space.
You may look at The space of Schwarz–Klein spherical triangles and references there, or my survey
Metrics of constant positive curvature with
conic singularities.
To derive a parametrization of polygonal regions from Kapovich-Milson seems hopeless, since the condition of existence of an extension of a "piecewise-immersion" of the circle to an immersion of the disk is too complicated to handle. Gabrielov and I tried this for the much more complicated problem of the moduli spaces of spherical polygons, and this does not work.
(If you wish to try yourself, here is a relevant survey: V. Poénaru, Extension des immersions en codimension one (d’après S. Blank), Séminaire Bourbaki, 1968, no. 342.)
