Question: "My question is do we need this hypothesis for the construction of the relative Proj? I think only quasi-coherence of L is needed."
Answer: You are correct. You find a proof in
Grothendieck, Alexander; Dieudonné, Jean A.
Éléments de géométrie algébrique. I. (English) Zbl 0203.23301
Die Grundlehren der mathematischen Wissenschaften. 166. Berlin-Heidelberg-New York: Springer-Verlag. IX, 466 p. (1971).
Proposition 9.7.7. They prove that for any scheme $S$ and any quasi coherent $\mathcal{O}_S$-module $E$, it follows $G:=Grass_d(E)$ is a separated $S$-scheme. If $E$ is of finite type it follows $G$ is of finite type. In particular it follows $\mathbb{P}(E^*)$ is a separated $S$-scheme for any such $E$. I believe the techniques of proof can be adapted to your situation of a general sheaf $L:=\oplus_{d \geq 0}L_d$ generated by $L_1$ as $L_0$-module.
Note: There is an equlity
$$Hom_{\mathcal{O}_S-mod}(L_1, L) \cong Hom_{\mathcal{O}_S-alg}(Sym_{\mathcal{O}_S}^*(L_1),L)$$
and you get an exact sequence
$$0 \rightarrow I \rightarrow Sym_{\mathcal{O}_S}^*(L_1) \rightarrow L \rightarrow 0$$
and $Sym_{\mathcal{O}_S}^*(L_1)/I \cong L$. Hence there is a closed subscheme
$$V(I) \subseteq \mathbb{P}(L_1):=Proj(Sym_{\mathcal{O}_S}^*(L_1))$$
and $\mathbb{P}(L) \cong V(I) \subseteq \mathbb{P}(L_1)$
and since $\mathbb{P}(L_1)$ is a scheme it follows $\mathbb{P}(L)$ is a scheme.