Is it possible to use Bento for Mac as the database for a database-backed website?
Bento isn't well-suited for your needs, in my opinion. It's possible to export information from Bento into a variety of formats, including text files, Numbers, or Excel. But it doesn't have the hooks needed to be queried by a Website creation tool. As you mention, FileMaker Pro does have that ability. Here is their web publishing page.

This may be overkill for you, but if you're familiar with SQL databases, you might also take a look at Sequel Pro.