As noted in the comments, the question has a positive answer for tame Artin stacks (see e.g. [Ols12, Prop 6.1]) and also for non-tame Deligne–Mumford stacks (see [KV04, Lem. 2]). It is however also true in general. Throughout, let $\mathscr{X}$ be an algebraic stack with finite inertia and let $\pi\colon \mathscr{X}\to X$ denotes its coarse moduli space.
Proposition 1.
The map $\pi^*\colon \mathrm{Pic}(X)\to \mathrm{Pic}(\mathscr{X})$ is injective and if $\mathscr{X}$ is quasi-compact, then $\mathrm{coker}(\pi^*)$ has finite exponent, i.e., there exists a positive integer $n$ such that $\mathcal{L}^{n}\in \mathrm{Pic}(X)$ for every $\mathcal{L}\in \mathrm{Pic}(\mathscr{X})$.
For simplicity, assume that $\pi$ is of finite type (as is the case if $\mathscr{X}$ is of finite type over a noetherian base scheme) although this is not necessary for any of the results (the only property that is used is that $\mathscr{X}\to X$ is a universal homeomorphism and that invertible sheaves are trivial over semi-local rings etc).
Lemma 1. The functor $\pi^*\colon \mathbf{Pic}(X)\to \mathbf{Pic}(\mathscr{X})$ is fully faithful. In particular:
If $\mathcal{L}\in \mathrm{Pic}(X)$, then the adjunction map $\mathcal{L}\to\pi_*\pi^*\mathcal{L}$ is an isomorphism and the natural map $H^0(X,\mathcal{L})\to H^0(\mathscr{X},\pi^*\mathcal{L})$ is an isomorphism.
The natural map $\pi^*\colon \mathrm{Pic}(X)\to \mathrm{Pic}(\mathscr{X})$ is injective.
Moreover, a line bundle on $\mathscr{X}$ that is locally trivial on $X$ comes from $X$, that is:
- Let $g\colon X'\to X$ be faithfully flat and locally of finite presentation and let $f\colon \mathscr{X}':=\mathscr{X}\times_X X'\to \mathscr{X}$ denote the pull-back. If $\mathcal{L}\in \mathrm{Pic}(\mathscr{X})$ is such that $f^*\mathcal{L}$ is in the image of $\pi'^*\colon\mathrm{Pic}(X')\to \mathrm{Pic}(\mathscr{X}')$, then $\mathcal{L}\in \mathrm{Pic}(X)$.
Proof. Statement 1 follows immediately from the isomorphism $\mathcal{O}_X\to \pi_*\mathcal{O}_{\mathscr{X}}$.
For the other statements, let $\mathcal{L}\in \mathrm{Pic}(\mathscr{X})$ and identify $\mathcal{L}$ with a class $c_\mathcal{L}\in H^1(\mathscr{X},\mathcal{O}_{\mathscr{X}}^*)$. If $\mathcal{L}$ is in the image of $\pi^*$ or locally in its image, then there exists an fppf covering $g\colon X'\to X$ such that $f^*\mathcal{L}$ is trivial. This means that we can represent $c_\mathcal{L}$ by a Čech $1$-cocycle for $f\colon \mathscr{X}'\to \mathscr{X}$. But since $H^0(\mathscr{X}\times_X U,\mathcal{O}_{\mathscr{X}}^*)=H^0(U,\mathcal{O}_X^*)$ for any flat $U\to X$, this means that $c_\mathcal{L}$ is given by a Čech $1$-cocycle for the covering $X'\to X$ giving a unique class in $H^1(X,\mathcal{O}_X^*)$. QED
As mentioned in the comments, when $\mathscr{X}$ is tame, then 3. can be replaced with: $\mathcal{L}\in \mathrm{Pic}(\mathscr{X})$ is in the image of $\pi^*$ if and only if the restriction to the residual gerbe $\mathcal{L}|_{\mathscr{G}_x}$ is trivial for every $x\in |\mathscr{X}|$ (see [Alp13, Thm 10.3] or [Ols12, Prop 6.1]).
Lemma 2.
If there exists an algebraic space $Z$ and a finite morphism $p\colon Z\to \mathscr{X}$ such that $p_*\mathcal{O}_Z$ is locally free of rank $n$, then the cokernel of $\pi^*\colon \mathrm{Pic}(X)\to \mathrm{Pic}(\mathscr{X})$ is $n$-torsion.
Proof. If $\mathcal{L}\in\mathrm{Pic}(\mathscr{X})$, then $\mathcal{L}^{n}=N_p(p^*\mathcal{L})$ (the norm is defined and behaves as expected since $p$ is flat). Since $Z\to X$ is finite, we can trivialize $p^* \mathcal{L}$ étale-locally on $X$. This implies that the norm is trivial étale-locally on $X$, i.e., $\mathcal{L}^{n}$ is trivial étale-locally on $X$. The result follows from 3. in the previous lemma. QED
Proof of Proposition 1. There exist an étale covering $\{X'_i\to X\}_{i=1}^r$ such that $\mathscr{X}'_i:=\mathscr{X}\times_X X'_i$ admits a finite flat covering $Z_i\to \mathscr{X}'_i$ of some constant rank $n_i$ for every $i$. By the two lemmas above, the integer $n=\mathrm{lcm}(n_i)$ then kills every element in the cokernel of $\pi^*\colon \mathrm{Pic}(X)\to \mathrm{Pic}(\mathscr{X})$. QED
It is also simple to prove things like:
Proposition 2.
Let $f\colon \mathscr{X}'\to \mathscr{X}$ be a representable morphism and let $\pi'\colon \mathscr{X}'\to X'$ denote the coarse moduli space and $g\colon X'\to X$ the induced morphism between coarse moduli spaces. If $\mathscr{X}$ is quasi-compact and $\mathcal{L}\in \mathrm{Pic}(\mathscr{X}')$ is $f$-ample, then there exists an $n$ such that $\mathcal{L}^{n}=\pi'^*\mathcal{M}$ and $\mathcal{M}$ is $g$-ample.
Proof. The question is local on $X$ so we may assume that $X$ is affine and $\mathscr{X}$ admits a finite flat morphism $p\colon Z\to \mathscr{X}$ of constant rank $n$ with $Z$ affine. Let $p'\colon Z'\to \mathscr{X}'$ be the pull-back. Then $p'^*\mathcal{L}$ is ample. We have seen that $\mathcal{L}^{n}=\pi'^*\mathcal{M}$ for $\mathcal{M}\in \mathrm{Pic}(X')$. It is enough to show that sections of $\mathcal{M}^m=\mathcal{L}^{mn}$ for various $m$ defines a basis for the topology of $X'$. Thus let $U'\subseteq X'$ be an open subset and pick any $x'\in U'$. Since $\mathcal{L}|_{Z'}$ is ample, we may find $s\in \Gamma(Z',\mathcal{L}^m)$ such that $D(s)=\{s\neq 0\}$ is an open neighborhood of the preimage of $x'$ (which is finite) contained in the preimage of $U'$. Let $t=N_p(s)$. Then $t\in H^0(\mathscr{X}',\mathcal{L}^{mn})=H^0(X',\mathcal{M}^m)$. But $\mathscr{X}\setminus D(t)=p(Z\setminus D(s))$ so $D(t)=X\setminus \pi(p(Z\setminus D(s)))$ is an open neighborhood of $x'$ contained in $U'$. QED
Acknowledgments I am grateful for comments from Jarod Alper and Daniel Bergh.
References
[Alp13] Alper, J. Good moduli spaces for Artin stacks, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 6, 2349–2402.
[KV04] Kresch, A. and Vistoli, A. On coverings of Deligne–Mumford stacks and surjectivity of the Brauer map, Bull. London Math. Soc. 36 (2004), no. 2, 188–192.
[Ols12] Olsson, M. Integral models for moduli spaces of G-torsors, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 4, 1483–1549.
Best Answer
A few responses to different parts of your question.
In my experience, the phrase "field of moduli" doesn't usually refer to the function field of a coarse moduli space. Rather: the base change of your cover to Lbar corresponds to a point of M(Lbar), where M/L is the coarse moduli space. This point has a well-defined field of definition, which is by definition the field of moduli of your cover. The phrase "field of moduli" is usually used in distinction with "field of definition" -- if your cover is actually defined over L', then the field of moduli is certainly contained in L', but it may not be equal to L'. This phenomenon isn't restricted to Hurwitz spaces; there are abelian varieties over Qbar whose field of moduli is Q (that is, they correspond to points of A_g(Q)) but which don't descend to Q. This can only happen when g is even. Off the top of my head I don't remember a reference for an example, nor for the assertion of the previous sentence; maybe somebody can help me out in comments. Certainly when g=1 you don't have this problem; given a rational number j, there is an elliptic curve E/Q with j(E) = j. But you prove this by writing it down -- it's not completely obvious "by pure thought" that it should be so.
The most complete description of the Hurwitz stack (the moduli stack of finite covers with fixed combinatorial invariants) its associated coarse moduli space, etc., is in the Ph.D. thesis of Stefan Wewers, which is unfortunately not available online. However, the survey paper of Romagny and Wewers should give you most of what you need.