> > > > > What animals have a thyroid?

Vertebrates. Invertebrate chordates have an organ called the endostyle, which has some functional similarity and is very likely the evolutionary predecessor to the thyroid gland.

> Is iodine chemistry localized in a different organ in the ones that don’t?

Well... many aquatic invertebrates may rely mainly on exogenous thyroid hormones, so they don't need an organ for it. Other invertebrates may have their own endogenous production of thyroid hormones, but without a dedicated organ. Yet other invertebrates, particularly outside the bilaterians, may be less reliant on thyroid hormones in the first place, since they seem to lack an ortholog of the thyroid hormone receptor.

Here is an interesting review article about thyroid hormone signaling in invertebrates.