The axiom AB6 holds in any Grothendieck topos. The reason is the following:
Filtered colimits and limits of sheaves are computed objectwise. (Since the sheaf condition is a finite limit condition, and finite limits commute with both in Sets/Abelian groups)
Therefore the natural map $$ \text{colim}_{\prod_{J} I_j} ( \prod M_{ij} ) \rightarrow \prod_{j \in J} \text{colim}_{i \in I_j} M_{ij} $$ can be checked valuewise for all $x \in X$ $$ \text{colim}_{\prod_{J} I_j} ( \prod M_{ij}(x) ) \rightarrow \prod_{j \in J} \text{colim}_{i \in I_j} M_{ij}(x), $$ which is an isomorphism since AB6 holds in the category of sets / abelian groups.
(I should remark here that an infinite product of filtered categories is again filtered, which is not hard to show but crucial for this argument)
Remark: The axiom AB4* holds as well. The category of abelian sheaves is closed under all limits for formal reasons, which includes products.