It is difficult to exactly map ancient concepts to modern ones. On the one hand, this means that what Aristotle means when he says "soul" (or rather, what is generally translated as "soul") is not what we think of when we say "soul" --people have suggested that "vital principle" or "life force" is a better translation. (It's also possible that it is closer yet still to the Chinese concept of qi.) So the idea of the Aristotelian "soul" describing something close to the modern idea of the immune system is not as outlandish it may appear on first glance.
However, there are things that Aristotle discusses as aspects of the "soul" that don't match up well with the immune system. Part of the problem comes from the fact that he distinguishes three different levels of soul, corresponding to plants, animals and people. But even at the base level, the plant soul includes self-nourishment and reproduction, neither of which are functions of the immune system. Then, sense perception comes in at the animal level, and rational thought at the human level, neither of which are immune system functions.
In summary, it's quite possible that Aristotle would have thought of the immune system, had he understood what we mean by the term, as a part or as a function of the "soul" (or vital force), probably all the way down at the plant level. But it certainly wouldn't encompass the entire soul. (Similarly the idea of qi is intimately connected with maintaining health. But it goes far beyond notions of fighting off disease.)