Creating an integration with Constant Contact would work with any program that allows itself to be integrated with an API that uses REST api's. I am not familiar with Filemaker, so I am not sure if this is possible. We do not have an official integration already set up with this, and I am not familiar enough with the program to know if there is a third party integration for it.
If you do want to start creating an integration with this product, I would suggest taking a look at our API documentation here. You can also take a look at our Marketplace, where there may be developers who are familiar with both Constant Contact integration, as well as FileMaker integration.
If you have any quesitons please let me know.