With only a few walk-in clinics nearby, CVS MinuteClinic may be your best option for immediate care in Dunkirk. As long as you visit them during their business hours, you should be able to walk right into 10095 Ward Rd, Dunkirk Marketplace and be seen relatively quickly. The national average for patients waiting at an urgent care is between 15-30 minutes, so hopefully it shouldn't be too long of a wait.
If you need a 24-hour option, you can always visit the closest emergency room, of course, you will likely pay 3-5 times more and have a much longer wait. In the case of a medical emergency, call 911.