What is z-pack antibiotics ? It is my understanding that antibiotics should not be prescribed for bronchitis because they actually have no beneficial effect and if you are in good health otherwise, then you are better without. So, my guess is you are a smoker. One of the side effects of antibiotics in bronchitis patients is vomitting, and dizziness is also one. What are your other medical issues at this time ? Do you have a heart condition ?
I can't go against what you doctor has prescribed for you and you should not go back to the pharmacist for anything 'extra'. If you are vomitting I would discontinue the medication and contact your doctor again.