If a pet dies while in the vets care then yes they will dispose of the pet for you at no additional charge unless you want cremation or something else for the pet then there would be a charge for the cremation of the pet.
Also if you have the vet put the pet to sleep as in euthanizing the pet then they will also dispose of the pet for you.
Sometimes vets will also dispose of pets for you when you bring the dead pet to them but there will be a charge which can very from vet to vet.
Or you can just dig a hole in your yard and bury the pet yourself in the ground at least 3 feet deep.
You don't need to do anything special to bury a pet like you would a person and just digging a hole and laying the pet into the hole and burying it is good enough.