When you tried the ODBC drivers, did you test them? Did you try them with other applications like Excel to confirm that they are working? FileMaker generally works pretty well with ODBC if the drive DSN is set up properly as a System DSN.
Alternatively, you can use the FileMaker ODBC driver to have Access connect to a FIleMaker file and you can push data to FileMaker tables. Remember that Access has a whole bunch of numeric formats. They work best importing via ODBC into FileMaker as a Number field with no other validations such as field length. Usually when I first push things, there is a field or two that is a problem. So I test importing just a few fields at a time with a test set of records and make adjustments as necessary. Acess will require a primary key for every table and FileMaker does not. So plan on making sure you create a Primary Key field that is validated as a required field that is unique.