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

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.
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.