Does anyone know if it's possible to add the name of a saved search to the results field? For example if my saved search is named "Vendors" is it possible to get the results as the list of our vendors with names, etc. and also include a column that has the saved search and is "Vendors" ?

Just so I better understand...Are you just looking for a column in the results of the saved search to show the name of that same saved search? If that's all you're looking at doing, you can add a Formula (Text) field to your results and for the formula just enter the name of the search, enclosed in single quotes. For your case- 'Vendors'.