Depends on device. Past jet injectors or air gun injectors used in vaccinations were able to transmit very tiny amounts of material (such as blood) from one person to another, and so were able to transmit hepatitis viruses. Current jet injection devices, if approved by the fda, are supposed to prevent cross-contamination among patients (most likely by using disposable single-dose vaccines).
Very unlikely. If your doctor follows proper sterilization procedures you will not get hepatitis c or any other comunicable disease from receiving vaccinations, air gun or otherwise. If you have concerns about sterilization procedures, you should talk this over with your doctor.