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The answer is yes. Since smoothness is preserved by flat descent, $X$ is smooth over $Y$. This implies that it has sections locally for the étale topology.

No ground field is needed: $Y$ could be any scheme and $G$ any smooth $Y$-group scheme.

[EDIT] to answer a comment by Keerthi Madapusi Sampath: if you don't assume that $X$ is a scheme but only a sheaf on the fppf site of $Y$, the answer is still yes. In fact, since $X$ is fppf-locally isomorphic to $G$, a theorem of Artin implies that $X$ is an algebraic space. In other words, there is a scheme $X'$ and a morphism $X'\to X$ which is representable, surjective and étale. Now $X'$ must be smooth surjective over $Y$, hence has sections étale-locally, and so does $X$.

If $Y$ is the spectrum of a field, group schemes and torsors are always quasiprojective. For any $Y$, if $G$ is affine over $Y$, then so are all $G$-torsors (in particular they are schemes). On the other hand, Raynaud has constructed a regular two-dimensional (local?) scheme $Y$, an elliptic curve $E\to Y$ and an $E$-torsor which is not a scheme.

2 added 486 characters in body

The answer is yes. Since smoothness is preserved by flat descent, $X$ is smooth over $Y$. This implies that it has sections locally for the étale topology.

No ground field is needed: $Y$ could be any scheme and $G$ any smooth $Y$-group scheme.

[EDIT] to answer a comment by Keerthi Madapusi Sampath: if you don't assume that $X$ is a scheme but only a sheaf on the fppf site of $Y$, the answer is still yes. In fact, since $X$ is fppf-locally isomorphic to $G$, a theorem of Artin implies that $X$ is an algebraic space. In other words, there is a scheme $X'$ and a morphism $X'\to X$ which is representable, surjective and étale. Now $X'$ must be smooth surjective over $Y$, hence has sections étale-locally, and so does $X$.

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The answer is yes. Since smoothness is preserved by flat descent, $X$ is smooth over $Y$. This implies that it has sections locally for the étale topology.

No ground field is needed: $Y$ could be any scheme and $G$ any smooth $Y$-group scheme.