Mo Mohamad
It is important to notice the difference between Map Viewer and Explorer in using geocoders. While Map Viewer can use all available geocoders in your organization (where the order of locators in the drop-down reflects their order in Utility Services list) and will give you the option to choose which locator you want to use to search for an address, Explorer uses either World Geocoder (when not signed in) or the first locator on the organization's locators list (when signed in).
If your web map that is using a custom locator in ArcGIS Online is accessed in Explorer by a public user who is not logged in, they cannot access the custom geocoder through your organization, and therefore, the app uses ArcGIS World Geocoder by default and there is no way to impose using a custom geocoder.. The behavior is different for users who sign in to the org.
For users who sign in and access the map, Explorer uses the locator that is at the top of the list in organization's Utility Services settings.
Putting the custom geocoder at the top of this list, and having users sign in to their accounts (in the same organization) is the only way to use a custom geocoder in Explorer.
In case you are using a field from a feature layer on the map as the search source, although you won't see suggestions from the feature layer in Explorer (unlike the Map Viewer) the locations from the layer are actually returned in the results regardless even if you are not signed in or signed in to a different organization.
All the best,
MJ