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.
