There’s no specific name for vegetarians who ride animals. Tipically, vegetarians don’t eat land-based meat like beef, pork, chicken, etc. However; vegetarians who ride animals do eat fish,–if supported. For this instance, you visited a farm where a vegetarian who rides a cow on a farm where dairy cattle are raised, but only few cattle. Or a vegetarian is riding a horse on a horse farm. There are some vegetarians who ride any animal as their mode of traveling.
Everyday, we ride different animals in our imagination, or via the real world. But some people has been rushing to have their animals ridden now. But Before you ever start riding your animals… you need to slow down, and take your time when training your animals to be ridden. If you rush; your animal you want to ride may NOT budge!
I believe some vegetarians ride animals,–as long they don’t overwork them. Training animals to be ridden can take time, and some years/generations. For example; you trained your cow to be ridden on your cattle farm where you only raise cattle for dairy (exclusively for lacto vegetarians). You don’t rely on whips, nor bullhooks. But you only train them with extreme time. In my opinion… you have to get used with bonding with your animal when training it to be ridden via each day. Rushing your animal to be ridden is NOT recommended because, your animal you’ve want to ride may NOT able to correspond with your desired commands correctly… that can also cause your desired animal for riding to fail overtime.