In bacteria and archaea, viruses are the primary infectious agents, acting as virulent, often deadly pathogens. A form of adaptive immune defense known as CRISPR-Cas enables microbial cells to acquire immunity to viral pathogens by recognizing specific sequences encoded in viral genomes. The unique biology of this system results in evolutionary dynamics of host and viral diversity that cannot be fully explained by the traditional models used to describe microbe-virus coevolutionary dynamics. Here, we show how the CRISPR-mediated adaptive immune response of hosts to invading viruses facilitates the emergence of an evolutionary mode we call distributed immunity - the coexistence of multiple, equally-fit immune alleles among individuals in a microbial population. We use an eco-evolutionary modeling framework to quantify distributed immunity and demonstrate how it emerges and fluctuates in multi-strain communities of hosts and viruses as a consequence of CRISPR-induced coevolution under conditions of low viral mutation and high relative numbers of viral protospacers. We demonstrate that distributed immunity promotes sustained diversity and stability in host communities and decreased viral population density that can lead to viral extinction. We analyze sequence diversity of experimentally coevolving populations of Streptococcus thermophilus and their viruses where CRISPR-Cas is active, and find the rapid emergence of distributed immunity in the host population, demonstrating the importance of this emergent phenomenon in evolving microbial communities.