Yes seat belts do lock up when airbags deploy as long as the seat belts are working properly.
When you get into a crash that causes the airbags to deploy then the seat belt should lock up first and the airbag deploy next.
That happens or should happen within a second of each other if everything is working properly.
Your seat belt locks up to hold you in place and then the airbag deploys to protect you from too much injury from hitting your head on the dash etc.
The reason that seat belts lock up after a car crash is to help keep the passenger or driver or whoever is wearing the seat belt from going forward and then possibly hitting their head on the dash, windshield or back seat etc.
When you get into a crash the computer detects the sudden stopping and then activates the locking mechanism on the seat belts so that it holds you in place.
Without the seat belt locking up and holding you in place you would still go forward in the crash which would not be a good thing.
Around 85 percent of people who drive or ride in vehicles wear their seat belts.
Most people only wear their seat belts if they are in an area where they may be police around that would be looking out for people wearing their seat belts.
That goes for me as well.
In my town there are no cops around unless someone calls them so mostly I ride around my town not wearing a seat belt as I hate the seat belts.
But my kids are always in their child car seats or in their seat belts regardless.
They know that by law they have to wear their seat belts or at least the 7 year old does.
So they wear their seat belts even when I don't wear my seat belt.
If I'm on the highway then I wear my seat belt or when I'm in a larger town where cops are.
Seat belts can save lives but they don't always do that and sometimes the seat belt can trap you in a vehicle such as during a fire etc.