No. Microsoft deliberately decided not to allow Access to run on linux 🙂
If the database is not complex, consider using MySQL.
Also, you can't (as far as I know) use ASPs directly from linux either. Consider using the tool ASP2PHP to convert your scripts to PHP. Unfortunately, I have had mixed results with this tool, especially when converting badly-written ASP.
Summary: You should tell your client that it is not a trivial matter to migrate from one platform to another.