You are getting that error because you are literally trying to execute
DELETE FROM @localVar
It's not just plugging in the value of that parameter like you want it to. You'll need to use dynamic SQL to accomplish this, and then as Rob said, use SP_EXECUTE.
DECLARE @localVar varchar(20);
DECLARE @qString varchar(max)
....
BEGIN
set qString = 'DELETE FROM ' + @localVar
SP_EXECUTE(qString)
END
...