She told herself that she could actually become as tall as she wanted.
Why use " as tall as" rather than " such a tall person that"?
Such means "of the kind mentioned or implied."
If you followup a such X with a later that, then you're "undoing" the "implied" meaning of such, which is contradictory and confusing.
She told herself that she could actually become such a tall person. (Fine, we're referring to "a tall person" that was previously mentioned or implied)
She told herself that she could actually become such a tall person that she wanted. (Now sounds like there's a specific tall person the writer/speaker had in mind that the implication created by such didn't cover - doesn't work.)
Of course, if only she has an idea of how tall she wants, and we know she is less tall than that amount, as tall as she wanted expresses this perfectly without us having to know precisely how tall that is, or otherwise read her mind.