The axiom AB6 holds in any category of sheaves on a site that can be constructed using only finite coverings. The reason is the following:

If checking for finite covers suffices, then filtered colimits and limits of sheaves are computed objectwise. (Since the sheaf condition is then 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)

However, in the absence of this finiteness condition, AB6 might fail, as then the sheaf condition might involve infinite products, and infinite products don't commute with filtered colimits. Therefore a sheafification is required which destroys the argument.