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Have you checked all the settings that say anything about 'Vibrate' ?
Have you checked whether 'Haptic Feedback' is enabled? it should be.
Have you checked whether 'Power Save Mode' is enabled? it should be tried as turned Off, because power saving might prevent vibrating.
If you do a factory reset, it will wipe individual data off the phone, and it might just free up the Vibrate function too. Have you tried that? or, do you need instructions?
Also, before the factory reset, you should go into the settings in the Messaging app itself, and make sure Vibrate is enabled there, for all notifications and messages, etc.
If you do not have data valuable to you, try the factory reset.
If you cannot even get to the factory reset option (usually reached, on many models if not yours, by holding down the Volume Down button, then holding down the Power button as well to turn it On... and sometimes another button too, in certain models...), or if you would rather not try the factory reset because it will wipe the phone clean of all your data (backing it up to another device at this point would be a good idea anyway) then the phone would need to be either repaired or replaced.