There isn't one specific word for a bug-eating mammal, but the most accurate term is insectivore.
"Insectivore" can be confusing because it's also a taxonomic order (Insectivora) that includes hedgehogs, shrews, and moles. These animals primarily eat insects, though their diets may include other invertebrates and sometimes small vertebrates.
Other Terms You Might Use
Bug eater: Simple and straightforward.
Entomophagous: A scientific term meaning "insect-eating."
Insectivorous: An adjective describing an animal that eats insects.
The best term depends on the context and what you want to emphasize.