I was curious if someone theoretically took the nazarite vow would they be able to eat/drink any fermented foods such as yogurt, cheese, leavened bread - as long as there is nothing derived from grapes? Also, would sugar alcohols like erythritol be permissible?

ב"ה Shalom A Nazirite is only forbidden to that which comes from the grape. (Rambam Nezirut 5:1). This question is not totally theoretical. Now too long ago, a very pious Rabbi, Rabbi David Cohen zt"l, (1887- 1072) (See: https://en.wikipedia.org/wiki/David_Cohen_(rabbi) ) took upon himself the restrictions of a Nazarite as a practice and did not drink wine or cut his hair. He would make Kiddush on Shabbat eve on bread. For Havdala and for the 4 cups at the Seder he would use alcoholic apple cider. (See: Ish Ki Yafli, The Nazarite Rabbi in Hebrew, by Hila Walberstein, pgs. 209,217, 228. ) All the best