So this is "kommer att" instead of "ska" because it`s a prediction and not her own intention? When would it be "ska" in this case?
Yes, that's right. If you say Jag vill att hon ska uppnå sitt mål ('I want her to achieve her goal'), you are the one who has the intention, which also works. She herself can say Jag ska uppnå mitt mål ('I will achieve my goal'), then it's her intention, both these sentences are good.
I feel like I'm missing something very basic but in that sentence "Jag vill att hon ska uppnå sitt mål" what is 'ska' doing there?
In English, you can say things like 'I want X to + infinitive', (I want her to reach her goal) but Swedish doesn't work that way: the infinitive isn't enough when the action is supposed to be performed by someone else. Since it's something you want to happen, it's a) in the future, b) there's an expression of someone's will in it, so we normally use the modal auxiliary ska which carries both these nuances of meaning. Especially in the spoken language, we also sometimes use the simple present. E.g. Jag vill att du ringer när du kommer hem 'I want you to call when you get home'.
But just vill + infinitive only works when the subject is the same: hon vill nå sitt mål 'she wants to reach her goal', but jag vill att hon ska nå sitt mål for 'I want her to reach her goal'.