How to get Zelda's horse in BOTW?

Assuming you are talking about the Royal White Stallion, it is found near the Manhala Bridge on Safulla Hill. An NPC at the Outskirt Stable should have mentioned it when they gave you the subquest.

Usually it is found a little south of the memory statue there. Yet, I know that some players had problems finding it, so there may be indeed some weird bug involved.

Anyway, as a suggestion, IF you manage by pure luck to get the Compendium picture for the horse (should be possible if you had all materials slot full when getting the original free pictures at the lab) then I think the slate allows you to track it.

PS: final suggestion - check you didn't already got a white horse in the past without realizing and check it didn't die.