Learn Turkish: Raporu yönetmene okutayım mı? - Shall I have the director read the report?

Because okumak (to read) already has a direct object in this sentence: raporu (the report). When you use a causative verb that already has an accusative direct object, the person who is made to do the action (the causee) takes the dative case (-e or -a).

Could I say Raporu yönetmeni okutayım mı instead?

No, that would be confusing and incorrect. If you use the accusative case for both nouns, it sounds like you are having an unmentioned third person read the report and also read the director! To show that the director is the one doing the reading, they must take the dative case here.

What exactly does the -t do in okutayım?

It is the causative suffix. It changes oku (read) into okut (make read / have someone read). Turkish verbs ending in a vowel usually take -t to form this causative stem.

How does okutayım mı translate to 'Shall I have... read?'

The verb combines the causative stem okut with the optative suffix -a (using a buffer -y-) and the first-person singular ending -yım. By itself, okutayım means 'let me have (someone) read'. Adding the question particle mı turns it into asking for an opinion or permission, which translates naturally to 'Shall I...' in English.