Is this sentence as ambiguous in Dutch as it is in English? ie could it mean either, "she's not coming, because of him" and "she's coming but not because of him"?
The sentence is indeed ambiguous. I think it depends on the word stress. If you say "De vrouw KOMT niet vanwege hem" the stress is on her not coming while "De vrouw komt NIET vanwege hem" means that she is coming, but not because of him. The program's pronunciation could be more fluent, but I'd say that she is not coming.
"omdat" is a conjunction meaning "because," whereas "vanwege" is a preposition meaning "because of." Just as you would the English counterparts, use "because" when you want to link two clauses together and use "because of" in front of a noun or pronoun.
Hope this helps! ^_^
I know this is old, but just to be super clear: you can't use omdat/want in the same sentence , but you could use it in a similar sentence like "De vrouw komt niet omdat hij dat zegt" right? Vanwege is only used in sentences like "The world disrespects America because of Donald Trump" or "My apartment is a mess because of my roommate"?