Haha, interesting question. Haven't thought of that. In many fanarts, they have. I believe they also have belly buttons in the show, it's just not visible because their мех covers it. I mean, equines are mammals, therefore if we approach the Вопрос from an anatomical perspective, they must have a belly button. :)
Um no, Лошади and ponies are animals, non human Животные would be weird if they had one. Non human Животные are not attached to an umbilical cord, I've never seen any non animal be attached to one.