Yes, you can say "We need to delay/postpone the delivery by a day." To 'put something off ' also means to postpone it - ' I'll put it off until Monday', it's more casual and is used in speaking. Postpone or 'suspend' are more formal : meetings may be postponed i.e. delayed until a later time. Delay is used for unforeseen events e.g. the train is delayed i.e. is behind schedule.