Is it possible to die from an air embolism from the equipment at the dentist's office?

Yes but not likely: Yes but not likely. Since dentist use many instruments that use compressed air, great care is taken not to force air into tissue spaces or wounds. Air embolisms are highly unlikely blowing air on hard tissues like teeth but open extraction sites and tissue wounds require caution.

Anything is possible: My guess is that you likely have a better chance of being struck by lightning for the 2nd time, than dieing from an embolism caused by dental equipment. Now in specific surgical procedures in specific areas of the mouth, if an operative hand piece is used in place of a surgical hand piece, there is risk, however, even then it is not likely to be fatal. I'm unaware of any such fatalities.