The claim "Antioxidants can also be important in improving immune responses in dogs." is being made on this web page. They go on to say
Because antioxidants play a key role in minimizing damage to cells, such as those that make up the immune system, recent research examined the benefits of certain antioxidants on the immune system activity of dogs. The results of these studies confirmed that antioxidants are important in helping dogs maintain a healthy immune system.
Though they don't say exactly what the "recent research" is. Is this just a case of noting that if you have too few nutrients then you will be ill, so eating these nutrients "improves" your health?