A limit sketch $\mathcal S=(\mathcal A,L)$ consists of a small category $\mathcal{A}$, together with a set $L$ of cones in $\mathcal A$. A model (in the category of sets) of a limit sketch is a functor $\mathcal A\to\mathrm{Set}$ which sends every cone in $L$ to a limit cone. We obtain a category of models for a given sketch with natural transformations as morphisms. By Corollary 1.52 in "Locally presentable and accessible categories" by Adámek and Rosicky, a category is locally presentable iff it is equivalent to the category of models of a limit sketch.
On the other hand, by Theorem 1.46 in the same book a category is locally presentable iff it is a full reflective subcategory of a presheaf category (over a small category) which is closed under $\lambda$-filtered colimits for some regular cardinal $\lambda$. In particular, this is the case for a topos of sheaves over a small site, see Proposition 3.4.16 in "Handbook of categorical algebra 3" by Borceux.
Consequently, any Grothendieck topos is equivalent to a category of models of a limit sketch and thus specifies a limit sketch up to Morita equivalence. Conversely, can we (nontrivially) characterize those (equivalence classes of) limit sketches whose category of models is a Grothendieck topos?