What is the best career for a person with social anxiety problems?

Not answerable: Obviously, there are too many factors to consider when a person pursues a career path. But it may help you to know that people with social anxiety often function very well when they have a specific role, meaning they have standing, purpose & function in a work setting. That is especially true if their position/role is clear -eg, a uniform, a station, a nametag, etc, that makes it clear what they do.

Tx before decision: Rather than let your anxiety dictate your career, you may want to find a psychologist who specializes in structured cognitive behavioral therapy so you can learn to understand and control you anxiety instead of letting it control you. It may be easier to pick a career that avoids social contact but it may be more rewarding in the long run to learn skills to become anxiety free.