The Mobotix I have is a top of the range one (AUS$3000) as it has night vision and many advanced features, but lower range ones are still very good products.
You can indeed link it to your website or use it's own webpage on your site.
As an example, our one is here: 202.139.23.24:8000
They can be customised with recording, taking snapshots etc...
But if these prove too expensive, there are many other brands within your price range that will do what you need.