Yes, this is definitely possible since FileMakerPro (FMP) supports AppleScript.
I don't have/use FMP, so I can't offer any specific solutions, but there have been many FMP questions in this Forum that will probably help you. If you do this Google site search of the forum you will get a lot of hits:
https://www.google.com/search?q=site%3Aforum.keyboardmaestro.com%20Filemaker
Once you have the data in an AppleScript, or in KM, from FMP, there are many different ways to populate a web page with that data. One of the ways is using Set Browser Field to Text Action.
So I'd suggest you review the Google site search results and then see if it becomes clear to you how to use AppleScript and KM to automate your workflow.
If you still need help, post back here with whatever scripts and Macros you have, and ask specific questions.
Also, Please read:
Tip: How Do I Get The Best Answer in the Shortest Time?