Pullback of locally free sheaves is locally free
Solution 1:
Answer fully inspired from @Kreiser's comments.
Let $x\in X, y=f(x)$. Let $V \subseteq Y$ be an open neighborhood of $y$ and $U=f^{-1}(V)$.
$G$ is a locally free $\mathcal O_X$-module, hence $G|_{V}\cong \bigoplus\limits_{i=1}^n \mathcal O_Y|_V$.
Pullback commutes with direct sum hence $(f^*G)|_{U}\cong \bigoplus\limits_{i=1}^n f^*(\mathcal O_Y|_V) \cong \bigoplus\limits_{i=1}^n \mathcal O_X|_U $ in a neighborhood of $x$.