Learn Hardcore Greek: Χρειάζομαι επειγόντως την υπογραφή σου για το συμβόλαιο. - I urgently need your signature for the contract.

Χρειάζομαι means I need. In Modern Greek it’s a middle/passive-form verb (ends in -μαι), but in meaning it’s usually just an ordinary active idea: I need / I require.
You’ll see it listed in dictionaries as χρειάζομαι (not χρειάζω in this sense).

What case is την υπογραφή σου, and why?

It’s in the accusative because χρειάζομαι takes a direct object:

την υπογραφή = the signature (accusative singular, feminine)
σου = your (informal singular possessive)

So the structure is basically: I need + (direct object).

Why do we use την here? Could it be without an article?

την is the definite article the (feminine accusative singular). It’s used because the speaker typically means a specific signature: your signature (the one we’re waiting for).
You can say Χρειάζομαι επειγόντως υπογραφή but that sounds more like I urgently need a signature (any signature / the act of signing), and it can feel less natural depending on context.

How does possession work in την υπογραφή σου? Why is σου not changing?