FileMaker does have a Scripting Language. Look under the "Scripts" menu. Select "Manage Scripts" That brings up the Scripting Environment within FileMaker. For more info on that see the help
http://www.filemaker.com/help/13/fmp/en/html/create_script.13.1.html#1034729
You can connect to the Mailchimp API using FileMaker scripts. Here is a Modular FileMaker Module that does just that.
http://www.modularfilemaker.org/module/mailchimp/
HOWEVER...
Coming from a web development background, you might be more familiar with thinking of FileMaker like MySQL and using any of the more common web stack tools like
PHP, Node.js, python or ruby. You can connect to FileMaker with any of those scripting languages (if your database is running on a FileMaker server). FileMaker has an XML gateway that you can use to connect with any language that can do HTTP requests ( like the ones I listed above ).
For more info see this
http://www.filemaker.com/products/filemaker-server/resources.html
There you will find links to the various ways that you can connect to FileMaker Server, which include the PHP API, the XML publishing gateway, and ODBC