I've found it in [FGA explained, Kleiman, The Picard scheme], p. 262, Exercise 9.4.3.
A universal sheaf/Poincaré sheaf exists iff $\mathbf{Pic}_{X/S}$ represents $\mathrm{Pic}_{X/S}$ or if $f: \mathscr{A} \to S$ has a section.
Edit: This gives us a Poincaré bundle on $\mathscr{A} \times \mathbf{Pic}_{\mathscr{A}/S}$, but I need it on $\mathscr{A} \times \mathbf{Pic}^0_{\mathscr{A}/S}$!