I'd just use MySQL for the web db, simple quick, easy AND RELIABLE. You can still use ODBC to connect to Access for dumping the data to your backend db.
Access is great for quick and dirty prototyping. It's just that it's not very robust when it's sitting on some host's web server. Indeed, Access is not very robust at the best of time, but with regular maint it is fine on a LAN with 8-10 concurrent users. You can even stretch the envelope up to 20-30 if you build in plenty of error traps and fault tollerance. Subject it to the kind of traffic that a website can generate and you'll loose data left right and centre, and it will fall over with maddening frequency.
It is really about choosing the right tool for the job.