Sounds like it all depends on what their system can do rather than filemaker.
You can export data from filemaker into a wide variety of file formats, excel, tab, commas separated text are three of the most common formats used for moving data between systems. Chances are extremely good that their software can import from some or all of these.
If they are willing and able to import data from such a file, then you can simply export the data into a compatible format and send it to them. It they aren't/won't then good luck.
Does their system allow to submit data by HTTP GET or only by POST?
If the former, you could construct an URL from their form parameters to submit your data by way of a web viewer or by using the Open URL script step.