Is there some kind of plugin or service for any browser out there (Mac OS, Windows or Linux) that would let me see a website like a colorblind person would?
This website seems to do what you need.
If you don't want to depend on a web service, Chrometric is a free AIR-based browser that lets you toggle between different forms of color blindness. It also has the advantage of working in a live browser (such as WebKit), rather than for single URLs at a time.
There's also a colorblindness simulator add-on for Firefox.