An adverbial phrase meaning 'as soon as possible' or 'as quickly as possible.' Common in both spoken and written Japanese.
確認が終わり次第、なるべく早くご連絡ください。
Please contact me as soon as possible once the check is complete.
なるべく早く会議の資料を準備します。
I'll prepare the meeting materials as quickly as possible.
都合がつけば、なるべく早く来てください。
If possible, please come as soon as you can.