Any auto-chef you buy in-game is ONLY for the restaurant it appears in. One single auto-chef does not continue through all other venues (restaurants) - as each one is designed uniquely and specifically for their "home" venue.

The Auto Chefs and their offers appear anywhere from the 1st to 2nd Seasons in most venues (it may take longer in later venues) - and are available to purchase once offered. It is permanently available to purchase once its offered with no expiration deadline.