What part of speech is 'stumble' in this sentence?

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.