The Search widget in Web AppBuilder for ArcGIS does not show suggestions when multiple search sources are set up and the display field returns a null record.
A best practice is to clean up the data so that the search field does not contain nulls. If the display field does not contain null records, it is fine to set up multiple search fields.Another option is to add the feature layer as a search source for each desired search field. In other words, avoid multiple search fields set up in an individual search source when there are null data.