It's important that a new theory of physics contains the equations and results of previous related ones. Maxwell theory and QED both have explanations for electromagnetic phenomena so I'm wondering if the Biot-Savart law can be derived in QED.
We can obtain Coulomb's law in the non-relativistic limit of the tree-level QED interaction, cf. this question. The Biot-Savart law is a consequence of Maxwell's equations, cf. this question. And Coulomb's law together with special relativity is sufficient to derive Maxwell's equation, cf. this question.
So, altogether, yes, we might say that we can derive the Biot-Savart law from QED.