Live, attenuated vaccines can potentially cause the disease they're meant to protect. One example is that the oral polio vaccine occasionally caused the polio virus. Others can cause virus symptoms: the chicken pox vaccine can occasionally cause a person to have a mild pox appear, and become contagious during that time.
Inactivated and synthetic vaccines cannot cause the disease they're meant to protect.
Although there is no real evidence that the thiomersal used in vaccines causes autism, vaccine manufacturers have been phasing it out of their vaccines for the last decade or so to stem controversy.