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 毛皮 covers it. I mean, equines are mammals, therefore if we approach the 質問 from an anatomical perspective, they must have a belly button. :)
Um no, 馬 and ponies are animals, non human 動物 would be weird if they had one. Non human 動物 are not attached to an umbilical cord, I've never seen any non animal be attached to one.