John Kruk was a great first baseman and outfielder. His lifetime batting average is .300. He was an All Star multiple times and played in a World Series. During an interview he was asked about being an athlete. He responded, “I’m not an athlete, I’m a baseball player.” Kidding aside, he was an athlete. Archers, too, are athletes though some may have a more Krukarian opinion of themselves.
Should you look at a training plan provided by USA Archery you’d note a section for weekly cardio/strength/conditioning sessions. While cardio is not exclusively running it remains exercise intended to develop cardio-fitness. Running is likely the first form of cardio that comes to mind.
Running is cheap. A pair of running shoes, which you probably already own, shorts and a t-shirt and you’re equipped for cardio.
Why do cardio? First off for your health. Look around at your next archery competition. You will notice a lot of overweight archers. Being overweight you know is not healthy. Secondly, as your fitness improves your heart becomes stronger. Your resting heart rate may lower. In archery a calm easy heart is better for shooting than a heart that is pounding away. Finally, if you are fit you may increase the years you have to live and thereby increasing the time you have to enjoy archery.
If you can’t run, due to poor fitness, you can walk. Start walking and see if it leads to running. If running simply isn’t your thing, there’s cycling. You might think about swimming. If you shoot a lot you’ll find swimming won’t give your deltoids and shoulders much of a break.
Archers are indeed athletes. As an athlete you need to consider the total picture of your health and fitness. Running or other cardio workouts might improve that picture.