Fresh Mozzarella with the rennet?
Close to 100% of cheeses require enzymes. Of those that don't, you can make a cottage cheese/paneer type thing, which is quick, easy and tasty:
(you can sub whole milk for the heavy cream
)you can also press it into a block to give it more density if you want.
and you can make fresh cultured cheese, but that's not quick--you have to let it ferment a bit. if you want to try it, see pg. 291 of anne mendelson's book Milk. too long to paraphrase! but there are probably similar recipes online.
ditto mascarpone--easy, no enzymes, but also takes time b/c it's cultured.
if animal products are in fact the issue, you can buy vegetable rennet on line. if you're into making cheese, try using rennet! it's not hard--there are plenty of non-aged cheeses like feta or queso fresco that you can easily make with rennet.