If you choose to do chemotherapy, this is important.
If you give too much antioxidents when using chemo drugs that are using an oxidizing type drug, (or radiation) it could possibly lessen the effect. (it's complex and controversial even among oncologists and many studies)
BUT not all chemo drugs work on a oxidizing basis.
Of course some use of antioxidents in any case would help minimize side effects and help the dog would tolerate it better. Double edge sword in these cases with some of those chemo drugs. But everything else I use would be important to support the dog.
Read these articles on the uses or non use of antioxidents during chemo and which drugs are problematic with antioxidents. This way you can make an actual informed choice, rather than get a blanket statement from a vet.