"Immune system" covers a lot of ground. The "immune system" that humans and other vertebrates have is a mishmash of evolutionary inventions going back well over half a billion years, that have been patched together with duct tape and bubblegum to form the rickety, clumsy and complicated apparatus that we know and love today.

Broadly speaking, the "immune system" gets divided into "innate" and "adaptive" branches. The "adaptive" part is what most people probably think of as "immunity:; it involved antibodies (B cells) and T cells. This arose in the ancestors of sharks, something like 400 million years ago.

Sharks and agnathans (jawless vertebrates, today represented by lampreys and hagfish) had a common ancestor maybe 550 million years ago, which did not have T and B cells as we know them, because lampreys and hagfish do not have T and B cells. (They do have their own analogies of antibodies, but they are a separate invention with very little in common with our antibodies.)

Although there have been some relatively minor changes in the vertebrate lineage, fundamentally we (and other mammals, fish, birds, reptiles) have the same immune system as sharks did.

The innate immune system is much older. We share recognizable components of innate immunity with insects. These include various pathways that recognize broad categories of pathogens, such as bacterial cell wall components. In vertebrates the innate immune system offers a large amount of rapid protection and also recruits adaptive immunity, which provides an initially slower but much longer-lasting protection.

Non-vertebrates don't have immune memory (you can't really vaccinate a bee, for example, though I know there are things that are rather sloppily called "vaccines") because "immune memory" is what the adaptive immune system does. Generalizing wildly, many invertebrates tend to be short-lived compared to vertebrates, which might make them less dependent on immune memory.

But of course there are very long-lived invertebrates too. In some cases invertebrates make much more extensive use of these innate immune components than vertebrates do. For example

>Detailed annotation of the Pacific oyster (Crassostrea gigas) genome, a protostome invertebrate, reveals large-scale duplication and divergence of multigene families encoding molecules that effect innate immunity.

--Massive expansion and functional divergence of innate immune genes in a protostome

Once you get much older than the vertebrate/insect split, immune molecules are harder to clearly identify. In bacteria, there are paths that protect against virus infection, but they are very different from the innate and adaptive immune molecules we see in eukaryotes. These include things like CRISPR and so on.