if you are ok with changing the language a little bit, then this is on p. 47 in this book http://www.math.ubc.ca/~cautis/dmodules/hottaetal.pdf plus Saito's Theorem 0.1 in MHM https://projecteuclid.org/euclid.pja/1195514125https://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=26&iss=2&rank=2.
Note that an $\mathcal{O}_X$-coherent sheaf is a D-module if and only it is a vector bundle with integrable connection. You can recover the D-module structure from your connection as follows: given $\mathscr{E} \to \Omega_X^1\otimes\mathscr{E}$ you can recover the D-module, i.e. the action of the tangent bundle via $\mathcal{T}_X\otimes \mathscr{E} \to \mathcal{T}_X\otimes\Omega_X^1\otimes\mathscr{E} \to \mathscr{E}$. For the sake of being pedantic, you can also check from Section 1.3 of the same book that for smooth morphisms, $f^*\scr{E}$ as a D-module will be the one coming from $(f^*\mathscr{E}, f^*\nabla)$. Now it should be the compatibility of de Rham fuctor with exterior product as in Theorem 0.1 of Saito. Perhaps for vector bundle with integrable connectoins Saito's theorem 0.1 was known before him?
As commented below this argument works for $S = \operatorname{Spec}\mathbb{C}$.