If a patient pulls away during administration of a vaccine and the needle comes out, is it okay to reintroduce the same needle and finish the injection?

No. The needle should be considered to be contaminated. The needle and syringe should be discarded. A new syringe, needle, and dose of vaccine should be used. Generally, a full repeat dose should be given, but you may use your clinical judgment to decide whether an adequate dose was administered before the patient pulled away.