What the achievement asks
Obtain fifteen of the twenty-eight Island Expeditions battle pets. Ten points, account-wide.
Buy them
This achievement has a reputation it no longer deserves. When islands were current, the pets dropped rarely, purchased pets did not count, and progress was locked to a single character. People ran eighty expeditions and saw one pet.
All three of those things were changed. Drop rates went up, the achievement became account-wide, and, decisively, pets bought from the auction house now count.
You do not have to farm this. Fifteen of these twenty-eight can be bought outright, and for anyone who is not specifically hunting pets that is the entire achievement.
Check what is listed and what it costs before committing to anything else. Prices vary enormously between species, and buying the cheap ten and farming the rest is usually better than either extreme.
If you do farm them
Two things are worth knowing.
The drop is not purely random: which pets can appear depends on what you kill, so an island held by spiders is the one that can give you a spiderling. Mythic difficulty does not improve the rate.
And when a pet does drop, it appears on the reward screen at the end of a successful expedition, not in the bag of goodies you open afterwards. It is easy to close through it without noticing.