Try using Firefox. I don't know if you can install the newest version but I bet they kept up Tiger support for a lot longer than Apple did so you would be at least a few more years advanced on Safari 4.1
The site does use Flash, at least for the things I viewed. I opened it with my new browser which does not have Flash and I saw black boxes where video appeared with my older browser with Flash.
A MacBook can be upgraded to at least OSX 10.6 and possibly even higher depending upon the exact specifications. You cannot upgrade to the newest OSX version and that is what they probably meant. Upgrading to a higher OS requires you purchase a 10.6 DVD which they do not sell in Apple Stores, only online.
1) Make sure your computer meets Snow Leopard technical specifications - http://support.apple.com/kb/SP575 This mostly means making sure it is an Intel Mac and has at least 1GB RAM (at least 2 GB is recommended). Go to the Apple in the upper left corner of any window to "About This Mac". To check RAM continue to "More Info...". It should say you have at least 1GB of "memory".
2) Buy a Snow Leopard disc from Apple online
http://store.apple.com/us/product/MC573/mac-os-x-106-snow-leopard - USA
http://store.apple.com/uk/product/MC573/mac-os-x-106-snow-leopard - UK
or telephone Apple. In the USA: (800) 676-2775 (contact information for other countries: http://support.apple.com/kb/HE57)
Alternatively you can buy from online retailers but anticipate paying much more than the original price which is the price still charged by Apple
3) Backing up your computer before upgrading is always a good idea.
4) After installing from the DVD, update your Snow Leopard installation using Software Update or manually downloading and installing the Mac OS X 10.6.8 Update Combo v.1.1 - http://support.apple.com/kb/DL1399
If you are using old (PPC generation) software you may want to install Rosetta as well which I believe is an optional installation.