Χρειάζομαι επειγόντως την υπογραφή σου για το συμβόλαιο.

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).

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).

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.

Greek often uses a possessive clitic after the noun:

• η υπογραφή μου/σου/του/της/μας/σας/τους = my/your/his/her/our/your/their signature

σου doesn’t change form for case or gender; instead, the noun + article change (here την υπογραφή).

Use σας instead of σου:

• Χρειάζομαι επειγόντως την υπογραφή σας για το συμβόλαιο.
  That’s the polite/formal your (also used for plural your).

επειγόντως means urgently / as a matter of urgency. It’s an adverb and is fairly flexible in position. Common options:

• Χρειάζομαι επειγόντως την υπογραφή σου… (very natural)
• Επειγόντως χρειάζομαι την υπογραφή σου… (more emphasis on urgently)
• Χρειάζομαι την υπογραφή σου επειγόντως… (also fine; emphasis can shift slightly)

για + accusative commonly expresses purpose / intended use:

• για το συμβόλαιο = for the contract (i.e., for the purpose of the contract / to finalize the contract)

If you wanted something closer to the contract’s signature, you’d normally rephrase, e.g.:

• Χρειάζομαι την υπογραφή σου στο συμβόλαιο. = I need your signature on the contract.

They highlight different ideas:

• για το συμβόλαιο = your signature is needed in relation to the contract / to proceed with the contract
• στο συμβόλαιο (σε + το → στο) = your signature is needed on the contract document (physical/locational sense)

Both can be correct; στο is more concrete.

Word order in Greek is more flexible than in English because the roles are marked by articles/cases. You can move parts for emphasis:

• Neutral: Χρειάζομαι επειγόντως την υπογραφή σου…
• Emphasis on your signature: Την υπογραφή σου χρειάζομαι επειγόντως… This fronting sounds more insistent: It’s your signature that I urgently need.

A practical pronunciation guide (approximate):

• Χρειάζομαι: khree-AH-zoh-meh (the χ is a “soft k/kh” sound, like German ch in Bach)
• επειγόντως: eh-pee-GON-dohs (stress on -γόν-)
• την υπογραφή: teen ee-poh-grah-FEE (stress on -φή)
• σου: soo
• για το: yah toh
• συμβόλαιο: seem-VOH-leh-oh (stress on -βό-)

την can drop the final -ν in certain contexts, especially before consonants (and it’s common in everyday writing too):

• την υπογραφή (before a vowel υ-, keeping -ν is standard)
• τη υπογραφή (you may still see/hear it, but keeping την is very common here)

As a learner, it’s safe to use την consistently at first; you’ll pick up -ν dropping patterns over time.