Hi, and welcome to Using English.
We do contract "want to" in our speech to "wanna." I want to go now. I want to see that movie.
It seems less likely that we'd contract in the way you've shown: "She's the one I want to see see succeed" or "Which does he want to be a doctor?" Possible, but not as likely.
Note that such contractions are not grammatical at all. It's simply a way of writing the sound of what we say. It's never grammatical to say "I wanna go to see that movie." It simply common to utter it that way.
I am among those who dislikes seeing wanna/gonna/etc. in anything but the MOST informal of writing platforms (text and e-mail to close friends), or as dialog in a novel, I cringe when I see "wanna" in an e-mail at work, and I am often so annoyed by it in posts here (except in cases like this when it's the subject of the inquiry) that I ignore the post.