You can store the file on Dropbox, yes. I've done that, as a matter of fact as a consultant working with clients.
However, there IS a risk if you all intend to work with that same relational database application sequentially. Discipline is crucial here. You can work on your local copy of the accdb if someone else is also working on their own local copy, but doing so will create data problems when you try to reconcile those two copies.
So, if you can maintain good discipline and avoid that risk, it can work.
That said, there are other, better ways to share data. One is to obtain a hosted SQL Server instance and migrate your data to it. Then, each of you can connect to that shared data and work as you wish. For as little as $5 a month you can obtain a hosted SQL Server or SQL Azure database. I believe that's going to work well for you here. And you can still use Dropbox to store a backup or master copy of the Access Front End accdb.