After thinking about this for some time after today's lecture, I believe the answer to your question is "yes" and I'll attempt to give a detailed proof (which might be a bit long for a post, but here we go).
Let $M$ be any $\mathbb C$-scheme and $h^M=\operatorname{Hom}_{\mathrm{Sch}/\mathbb C}(-,M)$ its represented functor. Let $\eta\colon \operatorname{Pic}_{\mathbb A^1/\mathbb C}\rightarrow h^M$ be any natural transformation; we must show that $\eta_X\colon \operatorname{Pic}_{\mathbb A^1/\mathbb C}(X)\rightarrow h^M(X)$ has image a single point for all $\mathbb C$-schemes $X$.
Step 0. We reduce to the case where $X$ is affine. Consider any affine open cover $X=\bigcup U_i$ and the commutative diagram
The right vertical morphism is injective because $h^M$ is a Zariski-sheaf. Thus it suffices to show that each $\eta_{U_i}\colon \operatorname{Pic}_{\mathbb A^1/\mathbb C}(U_i)\rightarrow h^M(U_i)$ has image a single point.
Step 1. We do the case where $X$ is reduced (and affine). For a point $x\in X$ let $\kappa(x)$ denote its residue field. We first claim that $h^M(X)\rightarrow \prod_{x\in X} h^M(\operatorname{Spec}\kappa(x))$ is injective again. After passing to an affine cover of $X$ that is mapped into an affine cover of $M$ and using some simple arguments as in Step 0, this boils down to the following question: Let $A$ be a reduced ring and $f,g\colon B\rightarrow A$ two ring morphisms that agree after composition with $A\rightarrow \kappa(\mathfrak p)$ for all $\mathfrak p\in \operatorname{Spec} A$. Then $f=g$. Indeed, the difference $f-g$ must have image contained in $\bigcap_{\mathfrak p\in \operatorname{Spec} A}\mathfrak p=0$ since $A$ is reduced.
Now consider the diagram
and observe that the bottom left product is a single point because both $\operatorname{Pic}(\operatorname{Spec} \kappa(x)[t])$ and $\operatorname{Pic}(\operatorname{Spec} \kappa(x))$ are trivial for all $x\in X$ (since line bundles over a UFD are trivial). So injectivity of the right vertical arrow does the trick.
Step 2. We do the case where $X$ is noetherian (and affine). To this end we claim $\operatorname{Pic}(X)\cong \operatorname{Pic}(X^{\mathrm{red}})$ for every affine noetherian scheme $X$, which immediately reduces everything to Step 1 [Edit: Turns out it doesn't, but fortunately Nuno has found a beautiful fix.] (this also uses $(X\times \mathbb A^1)^{\mathrm{red}}\cong X^{\mathrm{red}}\times\mathbb A^1$ which follows from a simple inspection). To prove the claim, let $\mathcal J$ be the coherent sheaf on $X$ cutting out its reduction $X^{\mathrm{red}}$. Since $X$ is noetherian, $\mathcal J^n=0$ for some $n$. Doing induction on $n$ we may assume $\mathcal J^2=0$. Now consider the short exact sequence $$1\longrightarrow (1+\mathcal J)\longrightarrow \mathcal O_X^\times \longrightarrow \mathcal O_{X^{\mathrm{red}}}^\times\longrightarrow 1$$ of multiplicative sheaves on $X$ (or rather its underlying topological space, which is the same as of $X^{\mathrm{red}}$). Using $\mathcal J^2=0$, it's straightforward to check that $1+\mathcal J$ is isomorphic to $\mathcal J$ (as an additive sheaf of abelian groups on $X$). Since $X$ is affine, $H^1(X,\mathcal J)=0=H^2(X,\mathcal J)$, so the long exact cohomology sequence provides the desired isomorphism $\operatorname{Pic}(X)\cong H^1(X,\mathcal O_X^\times)\cong H^1(X^{\mathrm{red}}, \mathcal O_{X^{\mathrm{red}}}^\times)\cong \operatorname{Pic}(X^{\mathrm{red}})$.
Step 3. We consider general affine $\mathbb C$-schemes $X$. Write $X=\lim X_\alpha$ as a cofiltered limit of noetherian affine $\mathbb C$-schemes $X_\alpha$ with affine transition maps. Using [Stacks project, Tag 01ZR & Tag 0B8W], one obtains $\operatorname{Pic}(X)\cong\operatorname{colim}\operatorname{Pic}(X_\alpha)$. The same holds for $X\times \mathbb A^1\cong \lim(X_\alpha\times\mathbb A^1)$, so actually $\operatorname{Pic}_{\mathbb A^1/\mathbb C}(X)\cong\operatorname{colim}\operatorname{Pic}_{\mathbb A^1/\mathbb C}(X_\alpha)$. Now every $\eta_{X_\alpha}\colon \operatorname{Pic}_{\mathbb A^1/\mathbb C}(X_\alpha)\rightarrow h^M(X_\alpha)\rightarrow h^M(X)$ has image a single point by Step 2, hence the same must be true for $\eta_X$ by the fact that the colimit in question is filtered. This finishes the proof.
I believe the argument in Step 3 can also be used (with some care) to show $\operatorname{Pic}(X)\cong \operatorname{Pic}(X^{\mathrm{red}})$ for arbitrary affine schemes, which would give an alternative to Step 3.
Best Answer
To start, here are some references. You can look at Conrad's notes as in Adeel's answer; the first page of those notes already contains what you are asking for in some sense.
Other references include the Stacks project, Vistoli's appendix to his landmark paper or the original paper of Deligne and Mumford. Of course, you can also look at the book of Laumon and Moret-Bailly.
Let me now try to explain a bit how to define the coarse moduli space.
Let $X$ be a finite type Deligne-Mumford algebraic stack of finite type over a noetherian scheme $S$ with finite diagonal.
In Remarque 3.19 of LMB (Laumon, Moret-Bailly) the coarse moduli sheaf of $X$ is defined to be the sheafification of the presheaf $$U\to \{ \mathrm{isomorphism \ classes \ of \ objects \ of \ X_U} \}. $$ This sheaf is not an algebraic space in general.
The paper of Keel and Mori proves that $X$ has a coarse moduli space. The coarse moduli space is defined as a morphism $X\to X^c$ of algebraic stacks with $X^c$ an algebraic space satisfying a certain universal property.
As Niels points out in the comments below and as mentioned above, the coarse space of a stack is not necessarily the coarse sheaf.
The representability of the coarse space $X_{coarse}$ of $X$ by a scheme is a difficult problem, as there are many algebraic spaces which are not representable by a scheme. I'm aware of essentially two non-trivial methods to see that $X_{coarse}$ is a scheme. These are GIT (Mumford) and the methods of Viehweg (see his book). It is with these methods that you can show that the coarse moduli space of the moduli stack of
smooth proper curves of genus $g$, or
principally polarized abelian varieties of fixed degree, or
canonically polarized varieties with fixed Hilbert polynomial, or
hypersurfaces of fixed degree $d\geq 3$ in $\mathbb P^n$ with $n\geq 3$, or
polarized K3 surfaces of fixed degree
is a scheme. (In some special cases you can also argue differently.)