"Writing" is a noun; "Write" is a verb. Nouns usually come after prepositions and "for" is a preposition, so the noun form is the only one that is appropriate in this context.

'Writing' is correct here. This is a gerund, and necessary in your sentence structure .

tips (noun) for writing emails (prepositional phrase that modifies the noun) The prepositional phrase here consists of a preposition "to", a gerund "writing", and an object "emails". Gerunds always functions as nouns so it fits in the structure of the prepositional phrase which is "to+noun".