Can juvederm reaction cause an allergic reaction?

Yes. It would be an extremely rare instance since the main component is hyaluronic acid which is present as a protein in your cells. However there is a rare instance that you are allergic. There is no allergy testing required for HA drugs.
Juvederm. There is the possibility of allergy and an allergic reaction when injected. Be sure to have your face evaluated by a board certified facial plastic surgeon and discuss the risks with your surgeon.
Possible... Short answer, yes. It is common for juvederm to cause swelling and redness at the site of injection. It is also common to have itching associated with it. These generally subside over time. If however you are having increasingly severe reactions to the injections with increased itching, swelling and redness each time you get the injection, you should consider an allergy.
It's rare, But possible. Best to return to the physician that injected you for an evaluation.