= Solution
{c}
Write $\mathbb P^r=\mathbb P^r_k$ over its ground <field>; the argument also works over any <Noetherian> affine base. Since $\iota$ is a <closed immersion>, $X$ is a <noetherian scheme> and a <quasi-compact scheme>. Let
$$
L=\iota^*\mathcal O_{\mathbb P^r}(1),\qquad F(n)=F\otimes_{\mathcal O_X}L^{\otimes n}.
$$
Here the <pullback of a sheaf of modules> makes $L$ an <invertible sheaf>, the restriction of the <twisting sheaf on projective space>. It is an <ample line bundle>.
By <global generation by sufficiently positive twists>, choose $n\geq0$ such that $F(n)$ is a <globally generated sheaf>. Finitely many <global sections> suffice: at each point, finite generation of the <stalk> of the <coherent sheaf> $F(n)$ lets us select finitely many generating germs of <global sections>. Those sections generate on a neighbourhood, because their <cokernel sheaf> is a <coherent sheaf> with zero <stalk> at that point. A finite subcover, using that $X$ is a <quasi-compact scheme>, supplies finitely many <global sections> generating everywhere. This is <finite global generation on a quasi-compact scheme>, and gives a surjective <sheaf morphism>
$$
\mathcal O_X^{\oplus m}\twoheadrightarrow F(n).
$$
Tensor with $(L^\vee)^{\otimes n}$, where $L^\vee$ is the <dual of a sheaf>. Here <tensoring with an invertible sheaf is exact>: locally the <tensor product of sheaves> is tensoring with a free module of rank one. Using the permitted identity $L\otimes L^\vee\cong\mathcal O_X$, we obtain
$$
\boxed{G=\bigl((L^\vee)^{\otimes n}\bigr)^{\oplus m}=\mathcal O_X(-n)^{\oplus m},\qquad G\twoheadrightarrow F\longrightarrow0.}
$$
A finite <direct sum of sheaves> of <invertible sheaves> is a <locally free sheaf> of finite rank, and it is a <coherent sheaf> on the <noetherian scheme> $X$. This constructs the required $G$. No regularity or smoothness assumption on $X$ is needed.
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