What the achievement asks
Collect thirty globes of Azerite Energy on Island Expeditions. Ten points, account-wide.
It is not the azerite you mine
This is the one that confuses people, because the island is full of azerite and only one kind counts.
Azerite Energy globes are dropped by enemies you kill. They appear on the ground where the enemy died, coloured like azerite, and you pick them up by walking over them. Mining azerite deposits, looting chests and gathering the ordinary azerite that fills the score bar do nothing at all for this criterion.
If your count is not moving, you are almost certainly mining rather than killing. Kill things and walk over what falls out of them.
Thirty is not many
Once you know what to look for, thirty globes takes only a handful of expeditions. The globes are not rare and not gated: they simply have to be noticed.
The habit that gets it done is walking through the spot where a pack died instead of moving straight to the next one. On a Mythic run where you are hunting elite packs anyway, that costs nothing.
Every expedition counts towards this, at any difficulty, so it will finish long before Expert Expeditioner or Azerite Admiral.