SearchContext can be bound both declaratively and programmatically server-side by setting the DataSource and DataSorceID property and assigning the DataTextField and DataKeyField properties to the desired fields of the DataSource so that the DataKeyField is the same as the one set to DataContextKeyField of Searchbox control.
In server side binding ImageUrl property can be assigned to the SearchContextItem in the ItemDataBound of SearchContext as it is achieved in the code in this article.
The items in the SearchContext drop-down can be also loaded from Web Service by setting the WebServiceSettings for SearchContext and the appropriate path and name. In this case one can assign the ImageUrl of an item directly in the service method as as shown in the following help article here.
When SearchContext is bound to RadODataDataSource control, its DataModelID property should be set to a Model ID from the data source. In such scenario in a similar manner as when the control is bound programatically, the DataContextKeyField field set to the SearchBox will be used as a context field. In order to set ImageUrl for the context items the OnClientItemDataBound event of the Search Context could be used as shown in this this article.