Add an accompanying Word Doc to my fanout zip folders

Yes this should be possible. You will need to use a Microsoft Word Writer set to fanout the same way that you are fanning out the shapefiles. I recommend referencing these for Word Writing:
https://docs.safe.com/fme/html/FME-Form-Documentation/FME-ReadersWriters/msword/msword.htm

and for working with zips:

https://docs.safe.com/fme/html/FME-Form-Documentation/FME-Form/Workbench/zip_files_writing.htm

Thanks--I’ve read up on both the writer and transformer but it seems like this is a way to create a word doc and not just read it and write it to a new location. However, it does say “The writer can either create a new document or append to the end of an existing base document” but, I don’t see a reader that allows me to bring in the Word doc.

Ah okay, I was thinking you were wanting to make new word docs during the process. If the files already exist and you want to copy them to new locations, then what you are looking for is the FileCopy Writer.