The web site cannot use the Adobe Reader application directly. You need to download the pdf and then you can view it with the Adobe Reader application. If that website doesn't allow the pdf to be downloaded and indeed requires a pdf reader plugin, off hand I don't think Adobe makes a mac browser pdf reader plugin. I could be wrong but I'm not aware of such a plugin.
On the other there does exist a pdf browser plugin. It's called, appropriately enough, "PDF Browser Plugin". You can find it here. Whether it works or not at that web site you are using is another question that probably can only be answered by trying it. The web site could be specifically checking for some Adobe plugin, in which case, this substitute will not work.
Are you sure you can't just download the file to use Adobe Reader, or for that matter, Apple's Preview app?