Definitely possible. One approach would be using NAV Webservices as the link on mibuso.com suggests and read/write NAV data real-time. An alternative (which we consider is better) - use replication approach, whereas the data you need to show on the website is replicated from NAV to MySQL, and the other way around. Replication will give you non-interdependent systems (i.e. the websites will still work while NAV is under maintenance or there's no link), improved performance of the websites, the ability to host them virtually anywhere, scale in the cloud etc., as there's no need to read anything from NAV while the visitors are just clicking. Supports older NAV versions as well.
An advice/tip - this is a complex ground. While you will surely create a proof-of-concept quickly, it will most likely take a big effort to to things right and reliable for serious production. We do NAV-ecommerce integrations on daily basis, integrating with PHP stack in particular [and there's a reason for that - PHP is around 80% share for data driven websites]. Anyways, initially it tool like 2 years and 10 projects of trial and error until honestly calling our integration engine stable and reliable.
Feel free to email for some futher advice. Or check out what we can offer as an ISV.