Both the phrasal verbs “use of” and “use by” are grammatically and semantically correct for that sentence construction, but the more conventional and idiomatic usage is “use of”: “This version is exclusively for use of students and teachers of (name of school).” I strongly recommend it over the “use by” variant, “This version is exclusively for use by students and teachers of (name of school),” particularly because of the much stronger association of the preposition “by” to authorship rather than to action in general.
Here’s how those two phrasal verbs work: In the prepositional phrase “for use of students and teachers,” the preposition “of” indicates the object (“students and teachers”) of an action denoted by the preceding noun (“use”). In “for use by students and teachers,” on the other hand, the preposition “by” indicates the agency or instrumentality (“students and teachers”) through which an action denoted by the preceding noun (“use”) is achieved. The grammatical mechanism of each of the two phrasal verbs differs, but the resulting semantics or sense is practically the same.
I trust that the above parsing of the two phrasal verbs has lessened rather than worsened your confusion over which of them is more appropriate to use.
