You can repeat the usual construction of a groupoid presentation of the Deligne-Mumford stack $\overline{\mathcal{M}}_{g,n}(X)$ from algebraic geometry, but instead using Douady spaces of compact complex analtyic spaces instead of Hilbert schemes of complex projective schemes.

Let $(X,J)$ be a compact, complex analytic space. Let $(C,p_1,\dots,p_n)$ be a projective, reduced, connected curve $C$ that is at worst nodal, and let $p_1,\dots,p_n$ be pairwise distinct, smooth points of $C$. For every embedding of $C$ into projective space such that the pullback of $\mathcal{O}(1)$ is sufficiently ample, an appropriate "slice" of the Hilbert scheme gives a *versal deformation space* of $(C,(p_1,\dots,p_n))$. This is a datum
$$
\xi = (\pi:\mathcal{C}\to B, (\sigma_i:B\to \mathcal{C})_{i=1,\dots,n}),
$$
of a projective, flat morphism $\pi$ of quasi-projective varieties -- both $\mathcal{C}$ and $B$ are smooth, as it turns out -- whose geometric fibers are reduced, connected, at-worst-nodal curves, together with an ordered $n$-tuple of sections $\sigma_i$ of $\pi$ whose images are disjoint and contained in the smooth / submersive locus of $\pi$. Moreover, $(C,p_1,\dots,p_n)$ is the fiber of the family over some $b\in B$, and the family is **versal**, i.e., for every $b'\in B$ with fiber $(C',p'_1,\dots,p'_n)$, the following Kodaira-Spencer map is surjective,
$$
\kappa_{b'}:T_{b'} B \to \text{Ext}^1_{\mathcal{O}_{C'}}(\Omega_{C'}(\underline{p}'_1+\dots+\underline{p}'_n),\mathcal{O}_{C'}).
$$
Since $B$ is quasi-projective, up to slicing $B$ by appropriate hyperplane sections, we can even assume that this family is **miniversal** at $b$, i.e., $\kappa_b$ is an isomorphism. Denote by $\omega_{\pi,\sigma}$ the invertible sheaf on $\mathcal{C}$,
$$
\omega_{\pi,\sigma} := \omega_{\pi}\left(\sum_{i=1}^n \underline{\text{Image}(\sigma_i)}\right).
$$

Form the product $\mathcal{C}\times X$ with the projection to $B$,
$$
\mathcal{C}\times X \xrightarrow{\text{pr}_1} \mathcal{C} \xrightarrow{\pi} B.
$$
This is a proper, flat morphism of complex analytic spaces. There is a relative Douady space of this morphism, i.e., a morphism of complex analytic spaces
$$
\rho:D \to B,
$$
and a closed analytic subspace of the fiber product, $$\iota:\mathcal{Z}\hookrightarrow D\times_B (\mathcal{C}\times X),$$ that is flat over $D$ and that is universal among pairs $(\rho,\iota)$ as above. The graph of $u$ gives a point in the fiber of $\rho$ over $b$. Near this point, the projection morphism,
$$
\mathcal{Z}\xrightarrow{\iota} D\times_B \mathcal{C}\times X \xrightarrow{\text{pr}_{1,2}} D\times_B \mathcal{C},
$$
is an isomorphism. For complex analytic spaces over $D$ that are flat, for a $D$-morphism of such complex analytic spaces, the property of being a local isomorphism is an open condition on the domain of the morphism. The closed complement is proper over $D$, so the image in $D$ is a closed analytic subspace $W$ of $D$. By hypothesis, $[u]$ is in the open complement $V = D\setminus W$.
Restricting over $V$, the projection morphism,
$$
\mathcal{Z}_V\to V\times_B \mathcal{C},
$$
is proper and a local isomorphism, i.e., it is a finite covering map. The degree of a covering map is constant on connected components. Thus, there is an open and closed subset $V_1$ of $V$ on which the degree of the covering map equals $1$, i.e., the covering map is an isomorphism of complex analytic spaces. Thus, $V_1$ is precisely the relative Hom complex analytic space, i.e., the restriction of $\mathcal{Z}$ over $V_1$ is the graph of a universal morphism,
$$
\phi:V_1\times_B \mathcal{C} \to X.
$$

Denote the base change over $V_1$ of $\pi$, resp. of $\sigma_i$, by
$$
\pi_{V_1} : \mathcal{C}_{V_1} \to V_1, \ \ \sigma_{V_1,i}:V_1\to \mathcal{C}_{V_1}, \ \ \mathcal{C}_{V_1} = V_1\times_B \mathcal{C}.
$$
The datum
$$
(\pi_{V_1}:\mathcal{C}_{V_1}\to V_1, (\sigma_{V_1,i}:V\to \mathcal{C}_{V_1})_{i=1,\dots,n}, \phi:\mathcal{C}_{V_1} \to X),
$$
is a family of prestable maps. Finally, for the pullback $\omega_{V_1,\pi,\sigma}$ of $\omega_{\pi,\sigma}$ to $\mathcal{C}_{V_1}$, there is a maximal open subset $M$ of $V_1$ over which $\omega_{V_1,\pi,\sigma}$ is ample relative to the morphism
$$
(\pi_{V_1},\phi):\mathcal{C}_{V_1} \to V_1\times X.
$$

Concretely, the union of all irreducible curves in fibers of $(\pi_{V_1},\phi)$ on which $\omega_{V_1,\pi,\sigma}$ has nonpositive degree is a closed analytic subset of $\mathcal{C}_{V_1}$ whose image in $V_1$ is also a closed analytic subset (since $\pi_{V_1}$ is proper), and $M$ is the open complement of this closed analytic subset.

The restriction of the family over $M$,
$$
\zeta_M = (\pi_M:\mathcal{C}_M\to M,(\sigma_{M,i}:M\to \mathcal{C}_M)_{i=1,\dots,n}, \phi_M:\mathcal{C}_M \to X),
$$
is a family of stable maps to $X$. By openness of ampleness, the open subset $M$ of $V$ contains the point $[u]$.

Since the family $(\pi,(\sigma_i))$ is miniversal at $b$, and by universality of $D$, the family $\zeta_M$ is a local analytic chart of the complex analytic stack $\overline{\mathcal{M}}_{g,n}(X)$ at $[u]$. For any versal family,
$$
\widetilde{\xi}= (\widetilde{\pi}:\widetilde{\mathcal{C}}\to \widetilde{B}, (\widetilde{\sigma}_i:\widetilde{B}\to \widetilde{\mathcal{C}})_{i=1\dots,n}),
$$
that is miniversal at the corresponding point $\widetilde{b}$ of $\widetilde{B}$, for the family $\widetilde{Z}_{\widetilde{M}}$ over $\widetilde{M}$ constructed as above, for the two pullback families over the product $M\times \widetilde{M}$, i.e., $\text{pr}_M^*\zeta_M$ and $\text{pr}_{\widetilde{M}}^* \widetilde{\zeta}_{\widetilde{M}}$, the relative Isom of the two families,
$$
I_{\zeta,\widetilde{\zeta}}=\text{Isom}_{M\times \widetilde{M}}(\text{pr}_M^*\zeta_M,\text{pr}_{\widetilde{M}}^* \widetilde{\zeta}_{\widetilde{M}}),
$$
is a Hom complex analytic space constructed by the same method as above (beginning with Douady spaces). Since the two families are families of stable maps and $\xi$, $\widetilde{\xi}$ are miniversal, also the two projections,
$$
q:I_{\zeta,\widetilde{\zeta}}\to M, \ \ \widetilde{q}:I\to \widetilde{M}
$$
are both local isomorphisms. Altogether, the family of "charts" $(M,\zeta)$ and the family of Isoms $I_{\zeta,\widetilde{\zeta}}$ form a groupoid presentation for a complex analytic (Deligne-Mumford) stack $\overline{\mathcal{M}}_{g,n}(X)$ in the category of complex analytic spaces.

When $X$ happens to be projective, the connected components of $D$ are each projective over $B$, and $B$ is quasi-projective, so that also $M$ is quasi-projective. Moreover, every Isom is also quasi-projective. Thus, when $X$ is projective, this even gives a groupoid presentation in the category of complex algebraic spaces.