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Have you tried to go in keyboard settings - sound and vibrate - see if vibration is ticked?
Can you boot in safe mode and see if the same occurs?
/t5/FAQ/How-to-boot-your-phone-into-safe-mode/m-p/348008#U348008
I just want to exclude any software cause.
Last shot before you have the device checked at a sony service is to repair software. You can do that or you can send it for repair.
Performing a software repair using PC Companion.The process will erase all internal memory.You might need to charge your device for about 1-2 hours before so the device has at least 60% battery.
Make sure the device is not connected to computer or laptop and turned off (sometimes you have to press power button and volume up until the device vibrates 3 times or on some devices you have to press the small button near the SIM tray or on the back of the device).
Open PC companion->support zone->phone/tablet software update->repair my phone/tablet blue highlighted.Follow the steps given,do not connect your phone until prompted to press volume down while plugging USB cable. If PC companion does not start the repair process, you might need to run it in compatibility mode with Windows 7 (in some cases XP).