At the moment there is no option in Star Chef to store, sell or delete Appliances - as each and every appliance you purchase has a purpose in your game.
Using your appliances to create dishes will provide the rewards (coins and experience points) that help you to complete cooking orders that help you progress through the game!
You can however store decorations in order to make room for more appliances, tables or other decorations. They will be kept safe and can be retrieved easily from storage through the decoration tab in the Shop. This way you can safely store and re-use those seasonal decorations, which are only available for a short amount of time in the game.