Blender isn't storing these normals for each shape key, so its not simply a matter of accessing some data.

While you could calculate them yourself, this isn't very convenient.

Temporary Mesh

• Pin the shape key so its active an no others are taken into account.
• Read the mesh using the Object.to_mesh method.
• Read the normals from the mesh.
• Remove the mesh.

Existing Mesh

• write a function that(optionally) store the vertex coordinates.
• temporarily appliesapply the shape-key to the mesh.
• calculate the normals.
• read them from the mesh (can store how you like, but be sure to copy them so as not to hold a reference to the original mesh data).
• restore the original coordinates and optionally re-calculate normals again
  (though this could be postponed until after you're done with all of the shape-keys).