ESRI Search Widget - Alter Search Parameters (ArcGIS JS v4.9)

The default source for the Search widget is the Esri World Geocoder. The suggest endpoint of the Geocoder does not allow this query format that you're asking about, so I don't think that's possible. If you have a Feature Service of your address points, you can do what you want by defining your own "source" to point the Search widget at your Feature Service, and then using esriConfig.request.interceptors to modify the request -- something like this:

esriConfig.request.interceptors.push({
  // set the `urls` property to the URL of the FeatureLayer so that this
  // interceptor only applies to requests made to the FeatureLayer URL
  urls: "https://ccmap.cccounty.us/arcgis/rest/services/LUCA/LUCA_ADDRESS/FeatureServer/0",
  // use the BeforeInterceptorCallback to check if the query of the
  // FeatureLayer has a where property set.
  // if so, then modify the "where" property how we want it to be.
  before: function(params) {
    if (params.requestOptions.query.where) {
      params.requestOptions.query.where = params.requestOptions.query.where.replace(
        "'%",
        "'"
      );
    }
  }
});

Full demo here.