Reservations can be made at all WDW restaurants, including Chef Mickey's, 180 days in advance. The best way is to book as close to the 180 days as possible, it does fill up quickly.

Many people get lucky just showing up and getting seated, but you take your chances, you may wait for a few hours and never get a seat (very likely). If you are doing this, either go very early or very late.

If you can't get a reservation, call dining from your room the day you arrive at WDW (or stop by guest services in one of the parks), they often hold cancellations for last minute reservations.