Is there a risk of anaphylaxis with apheresis?

Unlikely: "unlikely" because an aphorism in medicine is "never say never" but highly unlikely as the stuff returned to you is the stuff that was removed minus something. No foreign material is added to which your immune system could identify and react. In fact, apheresis aka plasmapheresis is sometimes used to treat serious cases of recurrent anaphylaxis where the cause is unknown or idiopathic.