I would split this problem up into two parts (here, 'sheaf (of groupoids)' is used instead of stack in order to disambiguate between Algebraic stacks (geometric objects) and mere (pseudo-)functors satisfying descent).  :

1.) Show that the inclusion of algebraic stacks into the category of fppf sheaves of groupoids on Sch preserves coproducts.  This follows immediately from the fact that algebraic stacks are a full (2,1)-subcategory of fppf sheaves and that if $F,G$ are two algebraic stacks, their sheafy coproduct is representable by an algebraic stack. If D is a diagram landing in a full subcategory whose limit or colimit exists in the ambient category and is in the full subcategory, this is also a limit or colimit of the diagram landing in the full subcategory without reference to the ambient category. I think the proof here is immediate by taking a disjoint union of the atlases.

2.) Show that colimits are universal in (2,1)-stack topoi.  This follows from the left-exactness of the stackification (2,1)-functor together with the altogether more obvious version of this fact for (2,1)-topoi of groupoid fibrations (also called (2,1)-presheaf topoi), where one can immediately reduce to proving the statement for groupoids pointwise.