Yes airbags do deploy in the event of a crash even when the seat belt is not on and not being worn.
The seat belt does not have to be buckled up or worn to allow the airbags in cars and any other vehicles to deploy.
Although not all crashes will causes airbags in vehicle to deploy but in a serious crash the airbag should deploy when the crash happens.
And having your seat belt on also helps to protect you even when the airbags deploy so make sure you have your seat belt on.
The airbags help to save lives as well but you can still be thrown out of the car or other vehicle in the event of a crash when you're not wearing your seat belt.
Also you can be slammed into the dash or windshield as well when not wearing a seat belt.
Another thing is if you're in the backseat of say a car or other vehicle and you're not wearing your seat belt you can be thrown to the front which happened to an aunt of mine over 25 years ago in a van.
My dad was driving and a car backed out of a school and my dad slammed on the brakes after he swerved to get away from that vehicle and my aunt went flying clear to the front and messed up her jaw but she was okay.
She was not wearing a seat belt and that is one reason you should be wearing a seat belt.