One common thought is that since exercise increases oxygen uptake from deeper breathing, extra antioxidants would be needed to stop the extra oxygen generated free radicals from damaging sore muscles while they attempt to recover and heal. You will hear this concept from many nutritionists and body builders. BUT, that is not what new research is showing. In fact, just the opposite may be the case. 
Extra antioxidant vitamins C & E may limit the benefits of exercise on muscle strength. Soreness after extreme exercising may be kicking up processes to heal and grow muscles, plus increase mitochondria for future greater energy production. ref
This next review looked at quite a lot of muscle soreness research, and didn't find quality studies proving either result. No benefits on relieving soreness but no adverse effects either, even though a few studies did find such results. This was just a review of other studies. Different dosages may be a problem factor on level of results.
This next study report looks at antioxidants on insulin sensitivity, a benefit from exercise. The antioxidant vitamins somewhat hindered this benefit. ref It turns out that the body might use the oxidation formed elements for some positive functions. Similar to how free radicals are formed by immune blood cells to help kill cold causing viruses and harmful bacteria. article
SUMMARY
It may very well be that while exercise increases oxygen radicals that could overpower the redox system, exercise also tells the body to generate more inside antioxidants to do the job. ref Plus, exercise stimulates appetite and more food is consumed supplying extra dietary antioxidants. Maybe these two processes are enough to neutralize the increased oxidation workload. ref Extra supplemental antioxidants might only be needed if other oxidation activities are also present. This is a likely scenario in play today. Thus, only small amounts of supplemental antioxidants may be all that is needed, especially since large amounts of antioxidant vitamins can interfere with redox balance. Thus the best antoxidant formulas are mostly food extracts and not just vitamins. See this article for more insight.
NOTE: Some people are just naturally low producers of ROS and don't require extra antioxidants while others produce high levels, likely driven by genetics, and would need extra antioxidant support. This would further be influenced by a good versus a poor diet in supplying needed nutrients, as well as exposure to environmental toxins, like lead, smoke, air pollution, cleaning chemicals, etc.
This is further proof that guidance is necessary to support the intelligent use of vitamins and dietary supplements. Vitamin Forms, Dosages, and Formula Composition are all vital.
FYI: Exercise is not an option. Body movement increases health benefits in numerous ways. A stronger heart muscle is at the core of protecting not only the cardiovascular system, but also the lymph circulation system which relies upon muscle movement actions for circulation. Increased digestion of nutrients is another way and don't forget about benefits to proper elimination.