It is possible, allergic reactions like anaphylaxis will set in between 10 and 15 minutes. These can be life-threatening if not treated immediately. The rate of anaphylaxis is 7 per million shots given.

That's why after receiving the vaccine you are supposed to wait 15 minutes so a medical professional can observe you for symptoms.

Not sure what happened in this case, but if it did happen it's likely someone botched the anaphylaxis response.