Children of Dune makes it explicitly clear that the sandworms are not native to Arrakis.
As for other life, the great likelihood is that almost all animal life is originally from Terra and then adapted, either through genetic artifice or more naturally over time, to the various worlds humans settled. Some species may be completely artificial, designed and cultivated in some lab here or there over the course of human expansion through space (I believe this is the provenance of the sandworm). The reason why this is probably true is two-fold:
- The incredible unlikelihood of animal species of a totally alien biochemistry evolving not only forms recognizable to human settlers but also of a composition that is useful to those same pioneers.
- The further unlikely situation that animal life would arise independently on multiple planets but not develop into intelligent forms either during the period of human exploration or at some time in the distant past and leave evidence of their existence.
Now, there are definite plant and bacterial-type species native to other planets which is making an interesting but implicit argument: abiogenesis is not unique to Earth, but the Great Filter lies at the line separating sessile, autotrophic, 'plant-like' life from mobile, voracious life. It would seem that in the Duniverse animal life either arose only once - on Earth - or it is extremely rare.
Finally, a very intriguing possibility is raised by the native flora of the planet Ecaz when asking if there's any intelligent alien life in Dune. Ecaz is home to fogwood, which can be molded by making skin-to-skin contact with the plant and simply thinking it into shape. Elacca comes from Ecaz, and this wood is the source of the drug which renders the user's consciousness totally sublimed to it, where physical self-preservation ceases to carry any weight. Elacca is also the source of semuta which also produces this 'consciousness-dispersing' effect. Ecaz is further the home planet of the sapho berry, which when processed into juice is drunk by mentats to increase their calculative abilities and sharpen their focus. Finally Ecaz is the source of Verite, the drug used by the BG and others as a truth serum.
What I mean by all that above is that life from Ecaz, always flora, interacts in powerful, meaningful ways with human mental faculties and processes. This may suggest that although animal life is uniquely Terran in the Dune series, consciousness and perhaps even intelligence has evolved at least one other time, in the various interconnected floral species of Ecaz.
In Dune's Terminology of the Imperium Frank Herbert tells us the sandtrout themselves are half-plant, half-animal, which might suggest an Ecazi origin as decomposers (sealing off the moisture and absorbing the nutrients of dead plant matter) before being unwittingly or short-sightedly carried to the planet of Arrakis when it was still a wet world. The Fremen keep predator fish in their caches so the sandtrout don't wall away all the water. That this works tells us the sandtrout not only didn't evolve on Arrakis, as they've no defense against the predator, but also that they seem to have evolved on a world with ample moisture but no animal predators. That could most certainly be Ecaz, especially when we consider what may have kept them from desiccating it was the conscious effort of the plant species that had evolved alongside them.