You could look into using WebDAV. This is what Sharepoint uses, for example, to achieve this functionality. It basically allows for a URL to act as a hard drive for the purpose of reading/writing files. Alternatively, you could just use Sharepoint for this functionality, but that would require a large amount of customization which isn't the easiest thing to do.
You could use a browser plug-in such as Silverlight or Flash. I have done this via Silverlight using COM interop with Word (as of SL5 this can be done in browser). I'm not sure if Flash has the same capabilities as I'm less familiar with it, but plug-ins have elevated privileges on the user's machine so it should be a lot easier to achieve this functionality.
You could create a client-side WPF application and have it installed on the user's machine via ClickOnce so that it is a fairly seamless experience. We have done this with one of our internal systems and it worked fine. Not sure how well it would work in a public-facing scenario, though.
Most of those options won't work in every browser on every platform. However, since you are targeting MS Word, I'm making an assumption that your goal is not to be 100% standards-compliant to get maximum reach. Most of these solutions will work for a majority of users on the major platforms.
Those options make the most sense if you are on top of a .NET stack already. There are many other potential options. If you give a better idea of what your current technology stack looks like you might get some more specific recommendations.