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We are interested in this as well. We have seen the forum post on getting form submissions into MySQL via PHP, but this seems like it could be problematic. If the PHP/MySQL server was to go down we would miss form submissions. We would much rather automate some type of "pull" or httpsget process.
The JotForm API makes it possible to connect to your form data without using the JotForm web site.
You can find more about different API http://api.jotform.com/docs/
Also I would recommend visiting developer forum http://developers.jotform.com/ if you any more questions about API.
Hope this will help.