Can you have a cavity filled without taking a shot?

Yup: But it is going to feel very sharp and sting and hurt. But I have had some people that want it done with out the shot...Some make it...Some don't.

That depends: That depends on the size and location of the cavity. Small - medium sized cavities on the chewing surface can often be fixed comfortably with out needing "a shot" (local anesthesia). Getting deep cavities and/or cavities between teeth fixed will definitely feel more comfortable if you do have local anesthesia placed.

Depends: on size of the cavity, your threshold for pain and your concerns about being numbed up. Talk to your dentist about your concerns.

Varies greatly: It's amazing the difference in pain threshold among patients. I have treated patients with giant cavities that feel nothing when I drill on the tooth, while others are so sensitive I can't properly even work on the tooth for an ordinary cleaning. Most are somewhere in the middle of that spectrum I often use nitrous oxide/oxygen analgesia (laughing gas=sweet air) to avoid using injection.