Xjyuk

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.

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.

If the player has the Smash Bros. Series Link Amiibo, you can use the
Amiibo Rune to tap Link onto the NFC reader. While most other Amiibo
will summon a supply drop of items - the first time the Smash. Bros
Series Link is used, Epona will be summoned instead.

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, if you wish to keep and ride Epona - you must register the
horse at a stable as soon as possible. If you summon Epona in a place
where you cannot get to a Stable, like on the Great Plateau, you will
be unable to summon Epona again, as using the Smash. Bros Series Link
Amiibo on other days will only summon standard supply drop items.

However; there is another article that mentions a fountain that may be of help to you.

The downside, of course, is that if Epona dies -- that's it. You
cannot respawn Epona by tapping the amiibo again after she dies, and
you will only be able to get her by restarting your game or by finding
the Horse God's Great Fairy Fountain.

So perhaps there is a chance you can get Epona back, based off of what that article says:

Unlocking this Great Fairy will allow you to bring any horse back to
life -- provided that it was registered, and that you have the means
to pay for it. Even legendary or rare horses can be brought back to
life at this fountain -- including Epona (provided you already
summoned her with an amiibo).

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.

It's very easy to miss on Epona if you use the Link Amiibo the first time in a place where horses cannot spawn. That may happen inside caverns, sanctuaries, ruins, on the volcano, or inside a village or a relay.

If you want to be sure, try a place where you see wild horses.

Note that you can only get Epone on the very first time you use the Amiibo in a given game. Trying again the next day, or after quitting and re-entering the game, or after using a different Amiibo, doesn't work.

The good news is that you don't have to restart the game from scratch if you miss Epona, provided that you act right away and load a previous save. Then you'll have another shot.

You're probably not using the right Amiibo.

I got Epona the first time I used the Super Smash Bros. Link Amiibo, that looks like Link with a Hylian shield in this battle pose. There is a chance you get treasure instead, but I think it's a 90% chance to get Epona.

Probably bad luck, or wrong Amiibo