How to receive Whatsapp notification without vibration? I am using Sony Xperia X, Android 6.0.1.

I have only managed to turn off the vibration of single and group chat messages by selecting "Off" under Vibrate and "Silent" under Notification tone.

The phone will still vibrate if I choose off under vibrate, and other than silent options under notification tone.

However, now without vibration, there isn't notification either whenever someone sends me a message. I am trying to receive message notification without vibration.

Such a simple permutation and I can't believe that I am still having trouble doing it.... OMG

Just tried on my Windows phone, same situation. Have to switch off whatsapp notification and vibration. Switching off vibration alone won't stop the phone from vibrating.
You can use Android's built-in permission framework appops. From adb shell execute command:

~$ appops set com.whatsapp VIBRATE deny