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Depending on the make, model and year of the car, the airbag module may shut off the fuel pump or disconnect a portion of the electrical system. Generally later model higher end cars will have this option and trigger it depending on the severity of the accident.
Unfortunately there is no real yes/no answer to the question since there is such a variety of cars and different passive safety strategies.
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This sounds like something you should report to NHTSA as a vehicle defect and then take it to the dealer for a field engineer to check the car over for the failure.
The link to NHTSA defects reporting is:
The link is self explanatory and can force a recall on the manufacturer if there are enough complaints.
The fact the side airbags deployed and the car did not shut down is probably a mix of faulty airbag module information and normal operation.