Is the pronoun in this sentence correct... Both the teachers and the students were delighted at him receiving such high SAT scores?

The pronoun in the sentence, 'him', is correct if the one the sentence refers to is a male. The pronoun 'him' is a singular, objective, personal pronoun which is functioning as the object of the preposition 'at'.

The preposition 'at' is not the best choice, a better choice is 'for' (were delighted for him).

Note: The Word 'both' can function as an indefinite pronoun when it takes the place of a noun. However, in this sentence, it functions as an adjective, describing the nouns 'teachers and students'.

Example use as a pronoun: The teachers and the students were both delighted...