I think this sentence could be read a number of different ways, but they all mean roughly the same thing. There is here a transitive verb, aperit (“opens” or “lays bare”), and a present active participle that could also take an object, ostendens (“showing”). There are a couple of accusative nouns, introitum and dolum (“entrance” and “artifice, contrivance” respectively), both of which equally fit as the object of either the verb or the participle. (introitum could be a perfect passive participle, but reading it that way makes the sentence needlessly complex.) Generally I’ve been taught that proximity is a good guideline for this sort of thing. Things attach to what’s closest to them.
As often happens, the best English translation entirely changes the word order of the Latin sentence. Latin usually plunks the verb down at the very end of the sentence whereas English generally wants the main verb as the second element in any sentence. My translation:
Daniel lays bare the artifice of the priests, showing the king their clandestine entrance.