Do I have a lawsuit against an auto manufacturer for airbags not deploying in an accident and suffering a broken jaw?
Yes, if the airbag does not deploy it MAY have been a manufacturing defect.
You'll need a lawyer, and proving the defect would take an expert who would examine the airbag, and then testify the defect was a result of something wrong in the design or manufacturing and not anything done afterwards, such as a mechanic disconnecting a triggering wire.
If it's just a broken jaw, and it heals without major long lasting damage, there may not be enough money in the case to make a major effort. BUT you may be one of many in the same situation and that makes a case very worthwhile.
Speak to a personal injury / products defects lawyer. You can find many really good ones at www.AttorneyPages.com