(I was hoping somebody else would answer this, because function fields are not really my area and I hoped I would learn something from the answer; but nobody seems to be biting, so...)
Iwasawa theory over function fields definitely exists, and in many ways it's easier than number-field Iwasawa theory -- there are more nice tools available, such as the Grothendieck--Lefschetz trace formula, which aren't there in the number field setting.
For instance, here is a paper of Goss and Sinnott from the 1980s which (among many other results) proves an analogue of Herbrand--Ribet for the class groups of function field extensions arising from Drinfeld modules.