When typing an address or a text in the search box, is there a way to stay in the current web map extent, and avoid going outside, in this case the County limits.
With portal, you can publish your own address locator that only includes locations within your county.
In AGOL, you can create a locator view from the ESRI World Geocoding Service.
When you create a locator view, you can define its extent in order to restrict search results to a specific area.
Once created, it is possible to share the view locator across to a Portal, but we have had instances where the view locator occasionally stopped working or sharing properly, which negatively impacted some of our public maps. Test it out, though!