Can I tell my antioxidants intake based on nutrition facts labels?
Well yes, you could get this information from food labels, since the recommended daily intake of vitamins is well established. But there is really no need to worry about dietary "antioxidants" (usually vitamins A, C and E, and carotenoids). If you eat a reasonably balanced diet, you are getting all the vitamins you need. Thankfully, vitamin deficiency is very rare in the western world today.

There is a lot of hype around "antioxidant" dietary supplements, but there is no convincing evidence that high levels of these vitamins in your diet has any beneficial effect at all. Randomized clinical trials have failed to show any effects whatsoever of antioxidant supplements on chronic diseases, and excess intake may even increase overall mortality.

Non-vitamin compounds like carnitine are not necessary in the diet at all, since they can be synthesized by the body in sufficient amounts. In general, the fact that a compound is a "major player" in cellular metabolism (in oxidative defense or other wise) does not mean that you have to consume it. For example, the TCA cycle intermediates are arguably the biggest "players" in human metabolism, but none of them are necessary in the diet.