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Let $Y (N) $ be the moduli scheme of dimension two principally polarized Abelian schemes with level $N$. It is claimed in "G.Laumon - Fonctions zeta des variétés de Siegel" (Lemma 4.1) that to an algebraic representation $W$ of $\mathrm{GSp}_{4}(\mathbb{Q})$ we can associate an $l$-adic smooth sheaf on $Y (N) [1/l] $.

Where can I find a proof of this please?

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This is a special case of Pink's "canonical construction" functor, which associates various kinds of coefficient sheaves on a Shimura variety (etale $\ell$-adic sheaves, vector bundles with connection, variations of Hodge structures, etc) to algebraic representations of the underlying group.

For more information see e.g. this paper by Torzewski: http://link.springer.com/article/10.1007/s00229-019-01150-9#Sec10.

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