Here is the full list of all 9 Food Delivery Battle achievements.
Hit the rocket against the other driver.
Megaphone Shockwave to hit other driver.
Drift and use a mini-turbo.
Become No.1 on the city stage.
Become No.1 on the planet stage.
Become No.1 on the lakeside stage.
Goal the time trial on the city stage within 40 sec.
Goal the time trial on the planet stage within 30 sec.
Goal the time trial on the lakeside stage within 23 sec.