Learn Italian: Iscrivetevi e portate i documenti che avete! - Sign up and bring the documents that you have!

Because iscriversi (to sign up / enroll) is a reflexive verb, and this sentence is an affirmative command. When giving an affirmative command informally (using the tu, noi, or voi forms), pronouns must attach directly to the end of the verb. Here, the voi command iscrivete (sign up) + the reflexive pronoun vi (yourselves) combines into the single word iscrivetevi.

The English translation just says "Sign up". Why does Italian use a reflexive verb here?

In Italian, the non-reflexive verb iscrivere means to enroll or sign someone else up (like a parent enrolling a child in school). To say that you are signing yourself up, you must use the reflexive form iscriversi. Literally, you are being told to "enroll yourselves".

Why doesn't portate (bring) have a pronoun attached to it like iscrivetevi does?

Because portare (to bring) is not being used reflexively in this sentence. The people are being told to bring the documents, not to bring themselves! Therefore, it just uses the standard voi command form: portate.

Is che required here, or can I leave it out like we often drop "that" in English?