From the PHB:
Beasts are nonhumanoid creatures that are a natural part of the
fantasy ecology. Some of them have magical powers, but most are
unintelligent and lack any society or language. Beasts include all
varieties of ordinary animals, dinosaurs, and giant versions of
Definition of Animal:
Animals are multicellular, eukaryotic organisms of the kingdom
Animalia (also called Metazoa).
The kingdom Animalia includes:
Animals are divided into various sub-groups, some of which are:
vertebrates; molluscs; arthropods; annelids; sponges; and
Arthropodae consists of:
millipedes, centipedes, insects, spiders, scorpions, crabs, lobsters, shrimp
Therefore the answer is "YES" a druid can become an ordinary bug or spider, or even earthworm.
An animal is basically any known living thing that is not a plant, a fungus, slime, mold or single cell organism or colonial establishment of single cell organisms. CAVEAT: I probably left something out of that what an animal is not list