Is it grammatically correct to say:
He is our chemistry teacher. He is our English teacher.
etc? Can academic subjects be used as adjectives?
Plenty of nouns can be used as adjectives; the grammatical term for this is noun adjunct, and it happens when a noun modifies another noun.
You can read all about them on the Wikipedia website (notice how I just used Wikipedia as an adjective there); other examples include: chicken soup, kitchen sink, light switch, brick house, biology teacher, and college professor.
Yes, that is correct. The position that the teacher holds is "chemistry teacher" or "English teacher", so referring to them as such is grammatically correct.