Does every relative curve have a Picard scheme? More precisely:

Let $X \to S$ be a smooth proper morphism of schemes such that the geometric fibers
  are integral curves of genus $g$.  Must the fppf relative Picard functor
  $\operatorname{\bf Pic}_{X/S}$ be representable by a scheme?

If $g \ne 1$, then some integer power of $\omega_{X/S}$ 
shows that $X \to S$ is projective 
(Remark 2 on p. 252 of 
Néron models by Bosch, Lütkebohmert, and Raynaud),
and then $\operatorname{\bf Pic}_{X/S}$ is a scheme by
Grothendieck, FGA, no. 232, Theorem 3.1.
So the question is really about the case
in which $g=1$ and $X \to S$ is not projective.
 A: The answer is yes: $\mathbf{Pic}_{X/S}$ is representable by a scheme. I will argue that this follows from the SGA 3 result mentioned by user27920 and from Theorem 2 (c) in section 6.6 of Neron models (which itself is based on a nonflat descent result due to Raynaud).
Preliminary reductions: Since the $g \neq 1$ case is clear, let us assume that $g = 1$. It suffices to work locally on $S$, so standard limit arguments reduce to the case when $S$ is Noetherian and affine. Since $\mathbf{Pic}_{X/S} = \bigsqcup_n \mathbf{Pic}^n_{X/S}$, it suffices to show each $\mathbf{Pic}^n_{X/S}$ is a scheme. The latter is a torsor under the elliptic curve $E:=\mathbf{Pic}^0_{X/S}$, so is an $S$-algebraic space of finite presentation. Further limit arguments therefore reduce to the case when $S$ is in addition local.
The main argument: We will argue that every finite set of points of $X$ is contained in an open affine. As explained by user27920, it will then follow from SGA 3, V.4.1 that each $\mathbf{Pic}^n_{X/S}$ is a scheme.
Choose an affine open $Y \subset X$ that meets the closed fiber of $X \rightarrow S$. Since $X \rightarrow S$ is open and $S$ is local, $Y$ automatically meets every fiber. The action map $E \times_S X \rightarrow X$ (gotten from the identification $X = \mathbf{Pic}^1_{X/S}$) is open because it is a base change of $E \rightarrow S$, so the image $EY$ of $E \times_S Y$ is open. Then checking fiberwise, we conclude that $EY = X$. Therefore, the claim about points being in an open affine follows from the result from Neron models mentioned earlier, a simplified version of which says:
Theorem. Let $S$ be an affine Noetherian scheme, let $E$ be a smooth $S$-group scheme with connected fibers, and let $E \times_S X \rightarrow X$ be a group action of $E$ on a smooth $S$-scheme $X$. If there is an affine open subscheme $Y \subset X$ such that $EY = X$, then any finite set of points of $X$ is contained in an open affine.
