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.