I'll try to negotiate with our accounting department/division about delivery date.
I'll try to negotiate with our accounting department/division to move up the delivery date.
negotiate with ~ about …
…について~に交渉する
accounting department/division 経理部/課
「納期について経理部/課に交渉してみます。」
move up the delivery date (forward~)
納品日を(~くらい)前倒しする
「納品日前倒しについて経理部/課に交渉してみます。」
例: move up the delivery days forward 2 days 2日前倒し