Can I send a Local Notification with no sound?

No you can not disable sound because UILocalNotification does not provide any option for this. So better option is as you told in your question to use a empty sound file.

Yes you can add another sound file.

NSString *soundFile=@"temp.mp3"; 

UILocalNotification* localNotification = [[UILocalNotification alloc] init];
    if (localNotification==nil) {
        return;
    }
    localNotification.fireDate = [NSDate dateWithTimeIntervalSinceNow:60];
    localNotification.alertBody = @"Your alert message";
    localNotification.soundName = soundFile;
    localNotification.timeZone = [NSTimeZone defaultTimeZone];

    [[UIApplication sharedApplication] scheduleLocalNotification:localNotification];

In above code just put the name of sound file which you have saved in your resources in place of "soundFile" string.