Quite a lot of supermarket Chedder in the UK is now vegetarian.

You can google this to learn more, but, my understanding of this is that the traditional way to produce cheese is to use rennet, a substance produced by young cows which helps them digest milk. As calves' stomachs are used, cheese made this way cannot be considered vegetarian.

I imagine that it's easier to standardise and produce large quantities of vegetable/fungal rennet type enzymes than use calf stomach in modern style mega-production so vegetarian cheese (in the UK at least) has become easier and easier to find in the last couple of decades, or perhaps sales do better when a manufacturer can show the little green vegetarian sign on the packaging. I couldn't say.

Kosher - I believe meat and milk can't be mixed in a kosher diet, so cheese made with meat/meat enzymes would be a no no. Milk curdled by vegetable rennet is probably fine.