Unless they are running an actual database (not a file-based system like MS Access, CSV, whatever) then connecting will be difficult to impossible.
If you can share over to the folder the database file is in, then you just have to setup permissions for the IUSR_
machine user to access the folder and file and write permission as well. Then you can use a normal connection string with a share path or mapped drive.
If they are running a real database, then a normal connection string should work just fine.
If neither of the above is possible, then you will need to either:
a) Supply a script/application for them to download and run
b) Make an ActiveX object
c) Create a form for them to upload the database, then change it, then send it back down to them
d) If it's MS Access, create a module in the database to use the XMLHTTP object everytime it is opened to open an ASP page on your server that dumps a list of SQL statements (basically a change log) to execute in order to get it up to date
e) If it's MS Access and your server has a real DB, create a linked table so that it will pull it's data back from your table
f) If both are MS Access force your user to download a new copy of the DB when you change the schema
-T