I could be wrong but I thought there was an add-on for this for a while although I don't recall the name or if it is even real or just something I imagined so maybe disregard this comment lol
I've been riding first person ever since I installed it and will never go back Lots of other camera tweak options in there too, like not changing to cinematic cams for NPC chats, so it's more like Skyrim - you can move the camera around during chats, stay in third person, etc. And account wide camera settings so no need to reconfigure everything for alts.
Animations for first person riding don't seem complete though (works well 90% of the time, but seems ZOS gave up towards the end), and quality varies with mount type. If you like camels... don't even attempt it. Wolves are amazing though