In this example, the verb 「奢{おご}る」could be used for buying someone something or treating someone to something. You could then express your sentence in the following way:
田中さんが本田さんを(私に)コーヒーを奢らせました。
The verb「奢る」works as follows: 「①が②に③を奢る」where ① is the treater, ② is the treatee and ③ is the treat. However, in the sentence above, there is no 「本田さんが」because because the subject of that sentence is actually 田中, not 本田.
But if 田中 is the subject, then how can the corresponding verb be「奢る」if he's not the one treating you? The verb form「奢らせる」actually means to make someone treat where the「せる」part is actually an auxiliary verb that does correspond with 田中 as the subject. You can find out more about it on this Japanese grammar website.
Note that this expression may sound as if 本田 was forced by 田中, therefore sounding a bit negative. I think that if you are the recipient of a treat, it's best to just stick with「本田さんにコーヒーを奢ってもらいました。」and not mention why he bought you coffee.