Yes, if it is completely essential then throw the exception. You should not* throw the exception later.
Always remember the “Fail Early Principle”. Concept being fail now, so you don’t waste time debugging or experience unexpected system functionality.
Alternatively you could also throw a ArgumentException for “” and ArgumentNullException for null. In either case make sure you throw a valid Exception message.
Always a good reference article for managing exceptions: Good Exception Management Rules of Thumb
Side note on what @Steve Michelotti said (because i am a huge fan of CodeContracts)
Contract.Requires<ArgumentNullException>(inputParemeter!= null, "inputparameter cannot be null");
Contract.Requires<ArgumentException>(inputParemeter!= "", "inputparameter cannot be empty string");
alternatively
Contract.Requires<ArgumentNullException>(!string.IsNullOrEmpty(inputParemeter), "inputparameter cannot be null or empty string");