Learn Hardcore Korean: siheom junbireul wihae jaryoga piryohaeyo. - I need materials to prepare for the exam.

In 시험 준비를 위해, 시험 준비 forms one unit meaning exam preparation. Here, 준비 is the head noun of that compound, so it takes the object particle 를. It’s similar to saying 공부를 위해 – meaning for studying.
If you said 시험을 준비하다, that would literally mean to prepare an exam (i.e. create the test). But when you want to express preparation for the exam, you use 시험 준비 and mark 준비 with 를.

What is the function of 위해 in 시험 준비를 위해?

위해 comes from the verb 위하다, which means to do something for the sake of. In the set phrase X를 위해(서), it functions as for X or in order to achieve X. So 시험 준비를 위해 literally means for the sake of exam preparation, i.e. to prepare for the exam.

Can I use 위해서 instead of 위해, and is there any difference?

Yes, both 위해 and 위해서 are correct. The only real difference is style/register:

위해 is slightly more formal or common in writing.
위해서 is often used in spoken or more casual contexts.
Both mean for or in order to, so you can say 시험 준비를 위해 자료가 필요해요 or 시험 준비를 위해서 자료가 필요해요 interchangeably.

Could I rephrase this as 시험 준비하기 위해 자료가 필요해요?

Absolutely. Instead of the noun 준비, you use the verb’s nominalized form 준비하기 plus 위해:
• 시험 준비하기 위해 자료가 필요해요
Both versions are natural and interchangeable. One uses a pure noun (준비); the other uses verb + -기 (준비하기).

Why is there no explicit subject like 저 or 제가 in this sentence?

Korean often omits the subject when it’s clear from context. Since you’re talking about your own needs, I (저) is understood. Including 저는 or 제가 자료가 필요해요 isn’t wrong, but dropping it makes the sentence more concise and natural.

Why is 자료 marked with the subject particle 가 instead of 를 when it’s the thing I need?

In Korean, 필요하다 is a descriptive verb meaning to be necessary. The thing that’s necessary becomes the subject. So you say:
• 자료가 필요해요 → Materials are necessary.
Even though in English we express it as I need materials, Korean literally says materials are necessary.

What exactly does 자료 refer to in this context?

자료 is a broad term for materials, resources, or data. In a study context, it usually means things like:
• Lecture notes
• Textbook handouts
• Practice questions or past exam papers
• Reference articles or online resources
Here, it simply means study materials.

Is 필요하다 a verb or an adjective, and how does 필요해요 function here?

필요하다 is considered a descriptive verb (often grouped with adjectives in English explanations). It describes a state of necessity. Conjugated in polite speech, it becomes 필요해요 (stem 필요하- + polite ending -아요/-어요). In 자료가 필요해요, you’re literally saying Materials are necessary. In English, we’d convey the same idea as I need materials.