How are vaccines mass-produced?
Per Wikipedia, typically when one needs a lot of virus, it is grown in a controlled cell environment. This used to be eggs, but is moving toward cell cultures instead.  So basically yes, factories full of virus (though more like in nice discrete bioreactors than big Joker-friendly vats).
Synthetic vaccines, such as the mRNA vaccines for COVID, do not need this step at all, since they do not actually use the virus, but can be done through cell-free biochemical reactions that replicate the mRNA directly.