When you say you don't want any monthly fees, are you thus expecting to find some database server system for free and without cost that allows external connections? I simply don't think you going to find such a service for free.
Since office 365 starts at $6 per month, then I not sure why you think this is too high of a cost here? So you could certainly consider to up-size your Access back end tables up to office 365 and continue to use your Access front end. And more amazing is with Access 2010 this means you get a "off line" and disconnected mode. This means that your application will continue to run EVEN WITH NO internet connection. The instant you find a wifi then the data sync process starts again. And this sync is not file based, but record based and is really replication built into the product and this setup requires ZERO extra code on your part.
And since your back end is not an access file, then you can now scale out to millions of users – the only real limit is the size of the 365 server farm (a super huge computer farm).
Keep in mind that in addition to simply linking your Access application to these office 365 tables, you can also publish Access web forms to office 365. So in the following video at the halfway point I switch to running the Access application 100% inn a browser:
http://www.youtube.com/watch?v=AU4mH0jPntI
Note the above resulting browser application does not require any ActiveX or Silverlight. And as noted, again this is again based on that massive server farm.
Another cloud approach is to consider SQL Azure. Access 2010 also has baked into the product the ability to use the cloud based edition of SQL server running on the Azure OS.
So, you could consider using SQL Azure, but that going to be about $10 per month.
I think the office 365 deal at $6 per month is the best bet (and you get lync communication which gives you remote desktop support for your customers or perhaps for supporting this application!). I actually think that Lync makes the $6 worth it alone. Toss in most SharePoint features and document sharing (including free web based Word, and Excel), this is hard to beat.
So it not clear here why you are avoiding office 365, but will have to adopt some kind of server setup here and I not aware of ANY system that going to allow free external connection from your desktop client software such as Access.
I think the best solution is 365 for use with Access.
Another low cost solution I used in the past is to consider some VERY low cost web hosting sites that also allow external connections to their database. In fact I did this for a good number of years (I did not even use the web site hosting!). I simply purchased the monthly web site and used the ability to "connect" external to the database server that was part of the web hosting package. This I did for a good number of years, and at VERY low cost. I thus was deploying Access front ends to multiple places and using this cheap-o web hosting account.
However, I am much now dropping this low cost web setup with the arrival of office 365 being even less cost than that cheap-o $9 web package I was using for this.
So, at the end of the day, I don't think there is any free hosting that allows external database connections, but the most low cost approach is office 365 at this point in time.
