Learn Portuguese (Portugal): Possivelmente, a trama revela o mistério da vingança. - Possibly, the plot reveals the mystery of the revenge.

You are right that doubt and possibility usually trigger the subjunctive in Portuguese! However, adverbs ending in -mente that express probability or possibility—like possivelmente and provavelmente—are special exceptions. They almost always take the indicative mood. If you swapped the adverb for an expression like talvez (maybe), you would then need the subjunctive: Talvez a trama revele...

Does possivelmente have to go at the beginning of the sentence?

Not strictly, but placing it at the beginning is the most natural way to frame the entire thought for the listener. You could also say A trama, possivelmente, revela... or put it at the very end, though moving it slightly changes the sentence's rhythm and emphasis.

Are there other ways to say "plot" besides trama?

Yes, enredo is another very common word for the plot of a story, movie, or play. The word trama can also translate to "scheme" or "conspiracy" depending on the context, but here it clearly refers to the storyline.

Why is it da vingança instead of just de vingança?

Using da (de + a) refers to a specific revenge—the one central to this particular plot. If you said um mistério de vingança, it would mean a general "revenge mystery", using the noun more like an adjective to describe the genre rather than pointing to a specific act of revenge.