Shiny Pet Charmer

You can obtain 8 Pet Charms from your garrison Pet Menagerie, per day.
Plus 4 Pet Charms from 6 different pet tamers in Draenor.
6 times 4 equals 24 Charms.
24 plus 8 charms equals 32 Charms.
500 divided by 32 equals 15 and a half days of Charm Farming.

Good Luck!

Since I've been doing this quest practically everyday for months and just noticed recently it hasn't tracked past 24 (since I've been spending them). I asked a GM, here's their response.

"Thanks for taking the time to contact us. This achievement will actually only track the number of Charms you have in your character's inventory, so while you may have looted hundreds in the past, 'Pet Charmer' will only count the 24 you have in your bags currently. I understand this can be a bit frustrating, but you'll just need to hoard all the Charms you receive until you get 500 - from there you'll receive the achievement."

This achievement is not contributed to by other characters on the account, so it is per character.

The Garrison pet daily, given by [npc=79858] (H) / [npc=85418] (A), awards 8 [item=116415] per day, for those who have unlocked [building=42] {Level 2 Garrison + Pets Versus Pests (H) / Pets Versus Pests (A)}.

If one has [building=168] unlocked, then completing Mastering the Menagerie (H) / Mastering the Menagerie (A) will award the [item=118697] which contains 8-10 [item=116415] (along with other things).

Along with the six (6) Draenor pet tamers who each award 4 [item=116415] for each win, the one garrison menagerie daily will yield a maximum daily total of:
24 charms (6 tamers x 4 charms) + 8 charms (daily) OR 8-10 charms (mastering) = 32-34 [item=116415] maximum, per character, per day.

Obviously, getting to a [building=168] a player will be well on their way to getting Shiny Pet Charmer, so I would not simply divide 500 charms by the 32-34 max daily earnings, but rather use it as a rough estimate of how many more days of grinding one has to do. For instance, I do not do the tamers at all, so I am earning a paltry 8-10 per day, so I have a long road ahead of me.