Yes you can run 32-bit apps under Vista x64. it uses the built in Wow64 layer, you don't need to install anything extra. But, not all applications will function as desgined, you may run into some issue. Applications that require a driver to be installed will not work as you cannot install 32-bit drivers on a 64-bit platform. As well programs that require kernel mode access, such as security related software will not function properly.
When you say "web-based" product, as the client are you just connecting through a web browser or is there client software to install? If it just requires a web browswer you should be ok, but the company may not provide you support if anything goes wrong.