Get the BBQ Machine in the Island Farm Store under the Machines tab. It becomes available once you reach at least 6000 Dev points.

You can and complete it with materials available from the Boat Ticket tab of your Mystery Store.

 The materials you can find in the Boat Ticket tab are related to your Boat Level:

- Seasoning Plate: Get this item from the Boat Ticket Store once you reached Wharf Level 6.

- Basket: Get this item from the Boat Ticket Store once you reached Wharf Level 7.

- Marinade Bowl: Get this item from the Boat Ticket Store once you reached Wharf Level 8.

- Brush: Get this item from the Boat Ticket Store once you reached Wharf Level 9.