According to Rav Kook, it is ideal to be a vegetarian. This is because he assumes G-d would not create a world in which creatures could not exist in harmony (see linked article) and therefore this would include killing animals. How, according to Rav Abraham Isaac Kook in A Vision of Vegetarianism and Peace would the necessary leather and parchment for things like Tefilin or Torah scrolls be obtained?
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Tefilin, etc., can be made from animals that die on their own (see e.g. The source in the Talmud is Shabbos 108a). Thus even under a situation where killing animals wasn't possible, leather would still be available.
You could ask the same question about sacrifices and egla arufa and many mitzvos that involve killing animals. Rav Kook's point is that killing/eating animals is a temporary allowance, and the ideal will return to the way it was with Adam who did not have the right to eat (and possibly to kill) animals, and did not have tefillin. For now, it is certainly allowed.
How he dealt with the Rishonim who say that there will be sacrifices in the third Temple, I don't know.