If it's a proposal that Section Chief Hida has vouched for, the client will probably approve it immediately.

Literal

Hida section-chief [subject-が] stamp-of-approval pushed proposal [conditional-なら] client [also-も] one-shot OK [probability-だろう].

太鼓判を押した is an idiom meaning 'to give a stamp of approval, to vouch for.' 企画書 means 'proposal/plan.' クライアント is a loanword for 'client.' 一発OK means 'approved on the first try.' Natural business slang.