Hi @phillip_bowman, you can begin this process by building your "query" using a combination of Tester or an AttributeCreator. If you already have a pre-built python or sql script that you were using you can integrate this into FME using the PythonCaller or SQLExecuter. Hope this helps you get started.
Here are an article that will help you get started with the process. Getting Started with FME Desktop: Perform a Spatial Filter & Modify Attributes
Not an expert,
But @phillip_bowman, @jennaatsafe is correct, I'd use tester to flick through your polygons and use spatial filters to search your layers. I've attached something I would use for your data set. I would use multiple spatial filters from one tester so each layer can be viewed at each stage then you can merge at output.