According to multiple sources online (including video evidence), you can get Epona in Breath of the Wild by using the Link Amiibo from Super Smash Bros.
Before knowing this, I had already used that Amiibo and it just gave me a few random resources. When I found out about Epona later, I assumed it probably didn't work for me because I did it too early (it was at the very beginning of the game, maybe even before getting to the first tower – the game probably wouldn't want me to have a horse at that point).
However, when I tried it again the next day (having made quite a bit of progress and owning three horses), the same thing happened again. Just random stuff, no Epona.
Are there any specific requirements for this to work, or did I somehow break my save file by using the Link Amiibo too early?
I'm playing the Wii U version if that makes any difference.
Best Answer
According to this article you should have gotten it on your first use of the amiibo. And if you did not put Epona into a stable you may not get her to summon again. It does not seem that there are any requirements for getting her to summon other than using the amiibo and it being the first time doing so, there are no indicators that using it at a specific point in the game changes how this works.
So my guess is, based off their explanation, is that you potentially summoned her and didn't notice and it may be stuck where you can't summon her again. Or you summoned her in the Plateau and you did not get to put her in a stable, thus she disappears.
However; there is another article that mentions a fountain that may be of help to you.
So perhaps there is a chance you can get Epona back, based off of what that article says:
But since it doesn't seem like you registered her if she did spawn by chance, it may not work either. The only solution remaining after that, as far as I can tell, is restarting, which I can't imagine you want to do.