Qualifications
The qualifications to become a cartographer are pretty simple. Earn a bachelor's degree (preferably in cartography, but if that's not possible, geography, geomatics, or surveying are all solid options, too), finish an internship, and boom. You're ready for a career.
Well...almost. There's still some licensing you may need to apply for, depending on your state. Some states require you to be licensed as surveyors (source), but it shouldn't be a problem to get yourself officiated.
There are other certifications you can receive to further your job opportunities as well, like the kind of stuff the American Society for Photogrammetry and Remote Sensing offers. But it's not required, just sometimes helpful if you're stuck in a jobless rut and looking for an extra nudge.