Why won't Laflat give me the "Lynel Safari" sidequest?
Solution 1:
You need to progress further on the main quest to get access to the Camera Rune.
After finishing the Great Plateau, you should've gotten a quest to go visit Impa called Seek Out Impa. Complete that quest, and do what Impa tells you to do there.
Impa will then give you a quest to go seek out Purah in Hateno Village called Locked Mementos. After meeting Purah, she will give you a task to go retrieve a blue flame. Complete this and she will give you the Camera Rune.