The usual argument is vaccines that use the WI-38 or MRC-5 cell lines which came from aborted fetal tissue. The Catholic church has official guidance on the six vaccines that were prepared using this line and indicates adherents have a duty to use vaccines from other sources. In the case that an alternate vaccine is not available, the church recommends fighting for the development of an alternative but draws the line short of demanding not getting vaccinated, suggesting that given two evils, the affected vaccine is the lesser.
The problematic vaccines are specifically the varicella (chicken pox) and rubella, for which there are no alternatives currently available. You have to work harder on the others because the affected rubella (and even varicella) components appear in many of the popular combination shots.
Still, certainly far from an outright prohibition.