If I was vacuuming chipped asbestos and my shop vacuum started blowing it into the air. Could I get sick?

Yes. You should always wear a respirator and protective clothing when handling asbestos. Additionally, the air when handling asbestos should be adequately filtered. Any aerosolization of asbestos particles can find their way into your lungs exposing them to asbestos fibers. That doesn't mean it will definitely lead to asbestosis, mesothelioma, or other lung cancers one day but why take the risk?