Here is a conditional answer to the question. Consider the following

Hypothesis: If $\mathcal Y$ is a sheaf $\infty$-topos, then for any set of objects $Y_0 \subset \mathcal Y$, there exists a small, full subcategory $Y \subseteq \mathcal Y$ such that $Y_0 \subseteq Y$ and a topology $J$ on $Y$ such that $\mathcal Y \simeq Sh(Y,J)$ is canonically equivalent to sheaves on the site $(Y,J)$.

• Note that the analog of this Hypothesis is true for 1-topoi, because for sufficiently large full subcategories we can always take sheaves with respect to the canonical topology.

• This approach to proving the Hypothesis doesn't work for $\infty$-topoi.

• In fact, I don't even know if the hypothesis is true when $\mathcal Y = Spaces$.

Claim: If the Hypothesis is true, then every $\infty$-topos is a sheaf $\infty$-topos.

Proof: Note that the classifying topos $\mathcal C$ for $\infty$-connective morphisms is a sheaf $\infty$-topos (see here for the proof that the classifying topos for $\infty$-connective objects is a sheaf $\infty$-topos). So is the classifying topos $\mathcal O$ for objects. There is a geometric moprhism $\mathcal O \to \mathcal C$ induced by a map of sites going the other direction.

If $\mathcal X$ is an $\infty$-topos, let $\mathcal X \to \mathcal Y$ be a geometric morphism exhibiting $\mathcal X$ as a cotopological localization of a sheaf $\infty$-topos $\mathcal Y$. So the localization is given by universally inverting some set $S$ of $\infty$-connective morphisms in $\mathcal Y$. By the Hypothesis, we may assume that these morphisms are between representables in the site presentation for $\mathcal Y$. So $\mathcal X$ is the pullback in the $\infty$-category of $\infty$-topoi $\mathcal X = \mathcal Y \times_{Psh(S) \otimes \mathcal C} (Psh(S) \otimes \mathcal O)$. The whole pullback diagram is induced by morphisms of sites, so the pullback $\mathcal X$ may be computed by taking a pushout in the $\infty$-category of sites and then passing to sheaves. Thus $\mathcal X$ is a sheaf $\infty$-topos.

Here is a conditional answer to the question. Consider the following

Hypothesis: If $\mathcal Y$ is a sheaf $\infty$-topos, then for any set of objects $Y_0 \subset \mathcal Y$, there exists a small, full subcategory $Y \subseteq \mathcal Y$ such that $Y_0 \subseteq Y$ and a topology $J$ on $Y$ such that $\mathcal Y \simeq Sh(Y,J)$ is canonically equivalent to sheaves on the site $(Y,J)$.

• Note that the analog of this Hypothesis is true for 1-topoi, because for sufficiently large full subcategories we can always take sheaves with respect to the canonical topology.

• This approach to proving the Hypothesis doesn't work for $\infty$-topoi.

• In fact, I don't even know if the hypothesis is true when $\mathcal Y = Spaces$.

Claim: If the Hypothesis is true, then every $\infty$-topos is a sheaf $\infty$-topos.

Proof: Note that the classifying topos $\mathcal C$ for $\infty$-connective morphisms is a sheaf $\infty$-topos (see here for the proof that the classifying topos for $\infty$-connective objects is a sheaf $\infty$-topos). So is the classifying topos $\mathcal O$ for objects. There is a geometric morphism $\mathcal O \to \mathcal C$ induced by a map of sites going the other direction.

If $\mathcal X$ is an $\infty$-topos, let $\mathcal X \to \mathcal Y$ be a geometric morphism exhibiting $\mathcal X$ as a cotopological localization of a sheaf $\infty$-topos $\mathcal Y$. So the localization is given by universally inverting some set $S$ of $\infty$-connective morphisms in $\mathcal Y$. By the Hypothesis, we may assume that these morphisms are between representables in the site presentation for $\mathcal Y$. So $\mathcal X$ is the pullback in the $\infty$-category of $\infty$-topoi $\mathcal X = \mathcal Y \times_{Psh(S) \otimes \mathcal C} (Psh(S) \otimes \mathcal O)$. The whole pullback diagram is induced by morphisms of sites, so the pullback $\mathcal X$ may be computed by taking a pushout in the $\infty$-category of sites and then passing to sheaves. Thus $\mathcal X$ is a sheaf $\infty$-topos.