Can a virus or bacteria get into the body through the ears? or is it only the eyes, mouth, and nose?

Infections: Different viruses & bacteria enter our bodies thru different pathways. In addition to what you've noted, don't forget sexually transmitted diseases eg genitals as another source of entry. To specifically answer your question, yes, bacteria can cause an infection of the outer ear aka swimmer's ear. Best way to avoid infections is to wash your hands. And get your vaccination status updated.