I saw this sentence somewhere: 'This episode sees the heroine stumble upon a body.'. I know 'stumble' is a verb, but which part of the verb is being used here? I don't think I really know how to parse any part of this sentence after the word 'heroine'.
This episode sees the heroine stumble upon a body.
"See" is a catenative verb, and the subordinate infinitival clause "stumble upon a body" is its catenative complement.
Syntactically "the heroine" is object of "see", but semantically it relates only to the subordinate stumble clause, not to "see". "The heroine" is thus called a 'raised object' because the verb it relates to syntactically is higher in the constituent structure than the one it relates to semantically.
The term 'catenative' comes from the Latin word for "chain", which is appropriate here since "see" and "stumble" do indeed form a chain of two verbs, in this case separated only by the NP "the heroine" and hence called a complex catenative construction.
In a comment, John Lawler wrote:
Yes. It's what's called an infinitive complement clause. It's a special variety because see is a sense verb, and they have unusual and extensive syntax. One of the specialities is that the ordinary infinitive marker to is omitted in clauses after non-volitional sense verbs (I saw him leave but not *I saw him to leave). The NP intervening between see and stumble is the subject of stumble, Raised to object position following see. This sentence could be parsed with the whole clause as object of see, or with the heroine as the object. Raising does both.