Hey @LouisCoetzee, thank for the question. Currently, there are two ways you can use an apiKey with the Search widget.
As of 4.18, you can use a global apiKey with esriConfig, which will work with the Search widget, and any other widget or basemap that requires an apiKey. Example based on your app:
https://codepen.io/noash/pen/yLgmRWK?editors=1000
At 4.19, you can also use a finely-grained apiKey on the LocatorSearchSource property of the Search widget. Example based on your app:
https://codepen.io/noash/pen/bGgXmZq?editors=1000
More information about apiKeys in general can be found here: https://developers.arcgis.com/javascript/latest/release-notes/#api-keys
