Do vegetarian athletes perform better?

No, vegetarian athletes do not perform better or worse compared to omnivorous athletes [1]. Studies have found that vegan, vegetarian, and omnivorous runners demonstrate comparable exercise capacity, suggesting that all three diets can be suitable for athletes with careful planning [2]. 

Although vegetarian diets aren’t necessarily superior for sports performance, vegetarian diets may have longer health benefits for athletes including reducing oxidative stress and regulating the anti-inflammatory response [1]. Plant-based diets may also be beneficial for improving cardiovascular markers, body composition, glycogen storage, and blood pressure [3]. 

That said, vegetarian athletes may be at increased risk of micronutrient deficiencies including omega-3, iron, zinc, calcium, and vitamin B12, and careful planning is required to ensure adequate intake of these nutrients and avoid decrements in performance [1]. For more information on plant-based diets for athletes see this article.