To get a Monster on Amber Island, you will first need to buy a Vessel from the Market using Relics. You can then revitalize the Vessel and transform it into an Egg by zapping the required Eggs to its Inventory. 

With the exception of Kayna's Vessel (which is always available), different Vessels are available at different times. Unlike Wublins and Celestials, the countdown timers for Vessels begin as soon as they are available for purchase in the Market. If you do not fill its Inventory before the timer runs out, the Eggs will expire and you will need to try again, but you will be compensated for the Eggs lost in the form of Relics.