With only a few walk-in clinics nearby, A - 1 Urgent Care may be your best option for immediate care in Key Largo. As long as you visit them during their business hours, you should be able to walk right into 101451 Overseas Hwy 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.