For a scheme $S$, every proper finitely presented map $f:X \rightarrow S$ from an algebraic space $X$ admitting a line bundle $L$ that is ample on each geometric fiber (which of course forces such fibers to be projective schemes, essentially by the very definition of ampleness for algebraic spaces) is automatically a scheme.

Indeed, by standard limit methods we may assume $S= {\rm{Spec}}(R)$ is local and then we assume only that $L$ is ample on the special fiber. That case can be descended to $S= {\rm{Spec}}(R)$ local noetherian, and the aim is to prove that for some large $n$ the power $L^n$ is generated by global sections with the resulting map $f:X \rightarrow {\mathbf{P}}(\Gamma(X,L^n))$ quasi-finite (so $X$ is a scheme, being quasi-finite and separated over a scheme). These are properties which are sufficient to check after faithfully flat base change to the completion of $R$, so we can assume $R$ is complete. It suffices to prove here that $X$ is now a scheme, as then the cohomological theory of ample line bundles would then apply to provide the desired $n$, the crux being the remarkable EGA IV$_3$, 9.6.4 (which has no flatness hypotheses).

Every infinitesimal fiber is a scheme, so we can make sense of the formal completion along the special fiber as a proper formal scheme equipped with a residually ample line bundle. By formal GAGA with the help of $L$, that formal scheme algebraizes to an actual proper scheme. But we have formal GAGA for algebraic spaces too, so the algebraic space algebraization and the scheme algebraization coincide.

QED