Artin's axioms do not apply in this case, because the stack is not limit-preserving. They only work with stacks that are locally finitely presented.
In any case, it is easy to give examples of quasi-coherent sheaves whose functor of automorphisms is not representable (for example, an infinite dimensional vector space), and this of course prevents the stack from being algebraic.
As Matthieu says, one should consider coherent sheaves.