どうして欲しいか私に言わなくてはいけません。
I'm guessing that I need to use something else besides the negative te-form + wa + ikemasen because that's more along the lines of "must do."
Also, when I have a verb in the -te form + hoshii, could I put hoshii all in hiragana?
なくてはいけません can be used both for "have to" and "must", or rather, it's hard to express the difference between "have to" and "must" in Japanese.
The point you must know is that どうして欲しい can have two different meanings and you seem to confuse them.
1)
どうして / 欲しいか
why you want them
This どうして is an adverb, meaning "why", not "どう + the -te form of する".
2)
どう / して / 欲しいか
how you want to be done
This is "どう(how/in what way) + the -te form of する". (Actually, this どうして is also usually treated as an adverb, meaning "in what way", but the structure you are misunderstood is equivalent to this interpretation.)
Thus, どうして is not "どう + して" in your translation.
You already mentioned that 貸して欲しければ in the first half of your sentence, so you don't need to repeat it. 私に is also unnecessary (the same reason as the conditional clause). It's OK just say 理由を(the reason) instead of どうして欲しいか私に.
Now you have an answer 私の本と鉛筆を貸して欲しければ、理由を言わなくてはいけません. See the difference between this answer and your initial translation あなたは、あなたに私の本と鉛筆お上げるのというが欲しければ、どうして欲しいか私に言わなくてはいけませんよ. It seems to me that this structure is advanced for you to translate yet.
By the way, this answer is grammatically perfect, but still sounds rather translationese-like, as already pointed out. I would just say どうして本と鉛筆を貸してほしいんですか instead in that situation.